Foundations of Algebra — Free Algebra 1 Review Games.

The commutative property of addition lets you swap the order of addends.

Distribute 3: 3*2x + 3*4 = 6x + 12.

Adding 0 to any number gives that number, so 0 is the additive identity.

|3 - 8| = |-5| = 5.

\(\sqrt{2}\) cannot be expressed as a fraction, so it is irrational.

4x - 2x - 6 = 2x - 6.

-3 is an integer and rational (can be written as -3/1), but not a whole number.

5x - 5 - 6x - 12 = -x - 17.

The multiplicative inverse (reciprocal) of 2/3 is 3/2 because (2/3)(3/2) = 1.

Without parentheses, exponent applies to 2 only: -(2^4) = -16.

This is the distributive property of multiplication over addition.

Inner: 3x - 4x + 4 = -x + 4. Then 2(-x+4) = -2x + 8.

The contrapositive negates and reverses both parts of the conditional.

\(3x^2 - x + 2 - x^2 - 3x + 5 = 2x^2 - 4x + 7\).

f(1) = 3(1)-2 = 1. f(f(1)) = f(1) = 1.

The multiplicative identity is 1 because multiplying any real number by 1 leaves it unchanged: a * 1 = a. Choice A (0) is the additive identity, not the multiplicative identity, since a + 0 = a. Choice C (-1) negates the number rather than preserving it.

The Associative Property of Multiplication states that the grouping of factors does not change the product: (a * b) * c = a * (b * c). Here the grouping changes but the order of factors stays the same. The Commutative Property (Choice A) would swap the order of factors, not regroup them.

The additive inverse of a number is the value that, when added to it, gives 0. Since -7 + 7 = 0, the additive inverse of -7 is 7. Choice B (-7) is the number itself. Choices C and D are the multiplicative inverses of 7 and -7, respectively, not the additive inverses.

First simplify inside the absolute value bars: -12 + 5 = -7. Then apply absolute value: |-7| = 7, since absolute value returns the non-negative distance from zero. Choice A (-7) is the value inside the bars before the absolute value is applied.

A rational number can be expressed as a fraction \(p/q\) where \(p\) and \(q\) are integers and \(q\) is not zero. \(0.75 = 3/4\), which satisfies this definition. \(\sqrt{3}\), \(\sqrt{7}\), and \(\pi\) are all irrational because they cannot be written as exact fractions and have non-repeating, non-terminating decimal expansions.

Using order of operations (PEMDAS): first evaluate the exponent: 3^2 = 9. Then multiply: 4 * 2 = 8. Finally add: 9 + 8 = 17. A common error is adding 9 and 4 before multiplying by 2, which incorrectly gives 26. Another error is computing (3 + 4) * 2^2 = 28.

Integers are the set {..., -2, -1, 0, 1, 2, ...}. The set {-2, 0, 3, 7} contains only integers. Choice A includes 1.5, which is not an integer. Choice C includes pi, which is irrational. Choice D includes 1/2, which is a fraction and not an integer.

Combine like terms separately: the x-terms give 3x - x = 2x, and the y-terms give 2y + 4y = 6y. The result is 2x + 6y. Choice B (4x + 6y) is incorrect because 3x - x = 2x, not 4x. Only terms with identical variable parts can be combined.

Apply order of operations: first the exponent: 2^3 = 8. Then division: 8 / 4 = 2. Then addition and subtraction left to right: 3 + 2 - 1 = 4. Choice B (5) results from computing 3 + 2 = 5 and then forgetting to subtract 1. Choice C results from subtracting 1 from 2 before adding 3.

The Associative Property of Multiplication states (a * b) * c = a * (b * c), allowing factors to be regrouped without changing the product. Here the grouping changes from 5 * (3 * x) to (5 * 3) * x. The Commutative Property (Choice A) would change the order of the factors, not just the grouping.

-3^2 means the exponent applies only to 3, so -3^2 = -(3^2) = -9. However, (-3)^2 means the entire quantity -3 is squared: (-3)^2 = 9. Therefore -9 + 9 = 0. A common error is treating both expressions identically and computing 9 + 9 = 18 (Choice B), ignoring the difference in how the negative sign is grouped.

Substitute x = -2 and y = 3: 2(-2)^2 - 3. Evaluate the exponent first: (-2)^2 = 4. Then multiply: 2 * 4 = 8. Finally subtract: 8 - 3 = 5. Choice B (-11) results from incorrectly computing 2x^2 as (2x)^2 = (-4)^2 = 16, then treating the coefficient incorrectly, or from distributing the square as 2 * (-2) * 2 = -8.

Distribute each factor: 2(3x - 1) = 6x - 2 and 4(x + 2) = 4x + 8. Then combine like terms: 6x + 4x = 10x and -2 + 8 = 6, giving 10x + 6. Choice B (10x - 6) results from distributing 4(x + 2) as 4x - 8 instead of 4x + 8, a sign error.

The closure property states that a set is closed under an operation if performing that operation on any two members always produces a member of the same set. For integers under addition, the sum of any two integers is always an integer. Choices A and C describe the commutative and associative properties, respectively. Choice D describes the identity property.

'Twice a number n' translates to 2n. 'Seven less than' that quantity means we subtract 7 from it, giving 2n - 7. Choice A (7 - 2n) reverses the subtraction, reading it as 'n doubled subtracted from 7.' Choice C (2(n - 7)) means 'twice the quantity of n minus seven,' which applies the multiplication after subtracting 7.

Apply order of operations: exponent first: 4^2 = 16, giving 16 / 16 * 2 + 3. Then multiply and divide left to right: 16 / 16 = 1, then 1 * 2 = 2. Finally add: 2 + 3 = 5. A common error (Choice D) is dividing 16 by 4 before applying the exponent, giving 4 * 2 + 3, then incorrectly continuing.

Natural numbers are {1, 2, 3, ...} and whole numbers are {0, 1, 2, 3, ...}. The only element in whole numbers that is not in natural numbers is 0. Choice A (-1) is in neither set, since both sets contain only non-negative or positive values. Choices C and D belong to both sets.

Distribute the negative sign across the parentheses: -(3x - 4) = -3x + 4. Note that the negative of -4 is +4. Then add 2x: -3x + 4 + 2x = -x + 4. Choice B (-x - 4) is the result of incorrectly distributing the negative sign as -(3x - 4) = -3x - 4, which fails to negate the subtracted term.

Work from the innermost grouping outward. Inside the brackets: 2x - (x + 4) = 2x - x - 4 = x - 4. Multiply by 3: 3(x - 4) = 3x - 12. Subtract 5x: 3x - 12 - 5x = -2x - 12. Choice D (4x - 12) results from adding 5x instead of subtracting it. Choice C (-8x - 12) results from distributing the 3 incorrectly.

p^2 is always positive regardless of the sign of p (squaring a negative yields a positive), and q > 0, so p^2 * q is the product of two positive quantities and is always positive. Choice A: negative times positive is always negative. Choice B: negative minus positive is always negative. Choice D: the sign of p + q depends on which term has the greater absolute value.

\(a^2\) is always greater than or equal to \(0\) for any real number, since squaring eliminates any negative sign and \(0^2 = 0\). Choice A is false: \(\sqrt{a^2} = |a|\), not \(a\) (e.g., \(a = -3\) gives \(\sqrt{9} = 3\), not \(-3\)). Choice B is false: \(|a| = a\) only when \(a \geq 0\). Choice D is false: \(-a < 0\) only when \(a > 0\) (if \(a\) is negative, \(-a\) is positive).

An expression is undefined when its denominator equals zero. For \(x / (x - 2)\): substituting \(x = 2\) gives \(2 / (2 - 2) = 2 / 0\), which is undefined. Choice A has denominator \(x = 2\), which is not zero. Choice C has denominator \(2(2) = 4\), not zero. Choice D has denominator \(x = 2\), also not zero.

Expand the left side: 3ax - 6 + 4x + 4b = (3a + 4)x + (4b - 6). Match coefficients with 7x + 2. For x-terms: 3a + 4 = 7, so 3a = 3 and a = 1. For constant terms: 4b - 6 = 2, so 4b = 8 and b = 2. Therefore a + b = 1 + 2 = 3.

Inside the parentheses, evaluate the exponent first: 4^2 = 16. Then subtract inside the parentheses: 16 - 6 = 10. Outside the parentheses, apply multiplication and division left to right: 3 * 10 = 30, then 30 / 5 = 6. Finally add: 2 + 6 = 8. Choice C reverses multiplication and division, but left-to-right order requires multiplying before dividing here.

A counterexample shows the claim is false. With \(a = 3\) and \(b = 4\): \(\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\), but \(a + b = 3 + 4 = 7\). Since \(5\) does not equal \(7\), this disproves the claim. Choices A and C both satisfy the equation accidentally (\(\sqrt{0 + 1} = 1 = 0 + 1\) and \(\sqrt{1 + 0} = 1 = 1 + 0\)), so they are not counterexamples.

The commutative property states that changing the order of addends does not change the sum: a + b = b + a. Here, 5 + 3 = 3 + 5 is a direct example. The associative property involves regrouping with parentheses, not swapping order.

An irrational number cannot be written as a ratio of two integers and has a non-repeating, non-terminating decimal. \(\sqrt{7} \approx 2.6457...\) is irrational. \(\sqrt{16} = 4\) is a rational integer, \(0.75 = 3/4\) is rational, and \(-3\) is a rational integer.

The coefficient is the numerical factor multiplied by the variable. In -7y, the coefficient is -7. The negative sign is part of the coefficient. Choosing 7 ignores the negative sign, which would produce an entirely different value.

A constant term has no variable factor. In 4x^2 - 3x + 9, only 9 contains no variable. The values 4 and -3 are coefficients attached to variable terms, not standalone constants in this context.

The associative property states that regrouping factors does not change the product: (a x b) x c = a x (b x c). Choice A shows the commutative property (changing order). Choice C shows the distributive property. Choice D shows the multiplicative identity property.

Absolute value represents a number's distance from zero on the number line, which is always non-negative. |-12| = 12 because -12 is exactly 12 units away from zero. Absolute value can never be negative, so -12 is an incorrect result.

Like terms must have the exact same variable raised to the exact same power. The term \(-2x^2\) has variable \(x\) raised to power \(2\), matching \(6x^2\). The term \(6x\) has exponent \(1\), \(x^3\) has exponent \(3\), and \(6y^2\) uses a different variable — none of these qualify as like terms.

Natural numbers are the positive counting numbers: {1, 2, 3, 4, ...}. The number 7 qualifies. Zero is a whole number but is not included in the natural numbers under the standard definition. Negative numbers and fractions are not natural numbers.

Following order of operations: evaluate the exponent first (4^2 = 16), then divide (16 / 2 = 8), then add and subtract left to right (3 + 8 - 1 = 10). A common error is adding 3 + 4 before squaring, yielding 7^2 / 2 - 1 = 23.5, which is incorrect.

Combine like terms: x-terms give 5x - 2x = 3x, and y-terms give 3y + y = 4y. The result is 3x + 4y. Unlike terms (x-terms and y-terms) cannot be combined. Multiplying the terms to get 6xy is an incorrect operation here.

The distributive property states a(b + c) = ab + ac. Here, 3(x + 5) = 3x + 3(5) = 3x + 15. The commutative property changes order, the associative property changes grouping, and the identity property involves multiplying by 1 — none of those apply here.

'Three times a number n' translates to 3n. 'Four more than' means add 4, giving 3n + 4. The expression 4n + 3 reverses the roles of 3 and 4. The expression 3(n + 4) means three times the quantity n plus four, which equals 3n + 12 — different from 3n + 4.

Substitute a = 3: (3)^3 - 2(3) + 1 = 27 - 6 + 1 = 22. A common error is computing 3^3 as 9 by mistaking it for 3 x 3 rather than 3 x 3 x 3 = 27, which would give 9 - 6 + 1 = 4, a result not even among the choices.

Rational numbers can be expressed as \(p/q\) where \(p\) and \(q\) are integers. In the third set: \(0.333... = 1/3\) (rational), \(-7\) is an integer (rational), and \(5/8\) is a fraction (rational). The other sets each contain at least one irrational number: \(\pi\), \(\sqrt{2}\), \(\sqrt{5}\), and \(\sqrt{3}\) are all irrational.

Distribute -2: -2(x - 4) = -2x + 8 (note that -2 times -4 equals +8). Then combine with 3x: -2x + 8 + 3x = x + 8. A common error is distributing as -2x - 8, forgetting that a negative multiplied by a negative gives a positive.

Applying the distributive property to 2(2x + 1) gives 4x + 2, which is algebraically identical to the first expression. Because the two expressions are identical after simplification, they are equal for every value of x — not just specific values.

Evaluate inside each absolute value first: -3 + (-2) = -5, so |-5| = 5. Then 5 - 8 = -3, so |-3| = 3. Finally, 5 - 3 = 2. A common error is evaluating |-3| + |-2| = 3 + 2 = 5 in the first term, incorrectly separating values before adding inside the absolute value bars.

Work from innermost groupings outward. Distribute: 3(x-2) = 3x - 6 and 2(x+1) = 2x + 2. Inside the brackets: (3x - 6) - (2x + 2) = x - 8. Multiply by 4: 4(x - 8) = 4x - 32. Add x: 4x - 32 + x = 5x - 32. A key error is writing -(2x+2) as -2x+2, forgetting to distribute the negative sign.

For any negative a, raising it to an odd power keeps the result negative: a^3 is negative (e.g., (-2)^3 = -8). In contrast, a^2 is always positive (even power), -a is always positive (the negative of a negative number is positive), and |a| is always positive by definition.

The first error occurs in Step 1. By order of operations, exponents must be evaluated before multiplication and addition. The correct sequence is: 4^2 = 16 first, then 3 x 16 = 48, then 2 + 48 = 50. The student incorrectly added 2 + 3 = 5 before resolving the exponent.

Factoring the numerator gives \((x+2)(x-2)\), so the fraction simplifies to \(x+2\) only when \(x \neq 2\). At \(x = 2\), the denominator equals zero, making the original expression undefined, while \(x + 2 = 4\) is perfectly defined. The expressions are therefore not equivalent at \(x = 2\).

Expand the left side: 2(3x + a) - (x + 4) = 6x + 2a - x - 4 = 5x + (2a - 4). For this to equal 5x + 8, set the constant parts equal: 2a - 4 = 8, so 2a = 12 and a = 6. A common error is setting 2a = 8 directly, ignoring the -4 that comes from distributing the subtraction.

Commutativity means a - b = b - a for all values. The counterexample 5 - 3 = 2 but 3 - 5 = -2 shows the results differ, disproving commutativity. Choice B demonstrates that subtraction is not associative, which is a different property. Choice C illustrates the identity property, and Choice D is a true algebraic identity, not a counterexample.

In -2^4, the exponent applies only to the base 2 (not to the negative sign), so -2^4 = -(2^4) = -16. In (-2)^4, the negative is inside the parentheses and is part of the base, so (-2)^4 = 16 (a negative base raised to an even power is positive). The sum is -16 + 16 = 0.

Distribute the negative sign: -(4 - x) = -4 + x. Distribute 2: 2(x - 3) = 2x - 6. Combine all terms: (-4 + x) + (2x - 6) = 3x - 10. A common error is writing -(4 - x) as -4 - x, incorrectly treating the negative of -x as -x instead of +x.

An irrational number cannot be expressed as a fraction p/q where p and q are integers. √7 ≈ 2.6457... is non-terminating and non-repeating, making it irrational. The other choices simplify to perfect squares: √9 = 3, √16 = 4, and √25 = 5, all of which are integers and therefore rational.

The identity property of addition states that adding 0 to any number leaves the number unchanged. Zero is called the additive identity. The commutative property would show n + 0 = 0 + n (order changes), and the inverse property would show n + (−n) = 0 (opposite values sum to zero).

The multiplicative inverse (reciprocal) of a number is what you multiply it by to get 1. Since (4/5) × (5/4) = 20/20 = 1, the multiplicative inverse of 4/5 is 5/4. Do not confuse this with the additive inverse, which is −4/5 and gives 0 when added to 4/5.

Natural numbers are the counting numbers {1, 2, 3, ...}. Whole numbers extend that set by including zero: {0, 1, 2, 3, ...}. Therefore, 0 is a whole number but not a natural number. −1 is an integer but not whole, and 1/2 and √2 are not integers at all.

A constant term is a number with no variable factor. In 7x² − 3x + 9, only the value 9 has no attached variable, so 9 is the constant term. The value 7 is the coefficient of x², and −3 is the coefficient of x. The answer choice 'x' is a variable, not a term at all.

The additive inverse of a number is the value that, when added to the original, gives 0. Since −5 + 5 = 0, the additive inverse of −5 is 5. Do not confuse this with the multiplicative inverse (reciprocal), which would be −1/5.

The associative property of addition states that the grouping of addends does not change the sum: (a + b) + c = a + (b + c). Choice A shows the commutative property, where order changes. Choice C shows the distributive property. Choice D shows the identity property of addition.

Follow the order of operations (PEMDAS). First, evaluate the exponent: 2³ = 8. Then divide: 8 ÷ 4 = 2. Then add and subtract left to right: 3 + 2 − 1 = 4. A common error is adding 3 + 2 = 5 before dividing, which is incorrect because division takes priority over addition.

The associative property of multiplication states that the grouping of factors does not affect the product: (a · b) · c = a · (b · c). Here the numbers stay in the same order but the parentheses — the grouping — changes. The commutative property would change the order of factors, such as 5 · 3 = 3 · 5.

Combine like terms by grouping the variable terms and the constant terms separately. Variable terms: 5x − 2x = 3x. Constant terms: 3 + 7 = 10. The simplified form is 3x + 10. Choice B (7x + 10) results from adding 5 + 2 instead of subtracting. Choice C (3x + 4) results from subtracting 7 − 3 instead of adding the constants.

'Twice a number n' translates to 2n. 'Five less than' that quantity means subtract 5 from it, giving 2n − 5. Choice A (5 − 2n) reverses the subtraction, changing the meaning. Choice C, 2(n − 5), means 'twice the quantity of n minus 5,' which simplifies to 2n − 10 — a different expression.

Substitute x = −2: (−2)² − 3(−2) + 2 = 4 + 6 + 2 = 12. Key steps: (−2)² = 4 (not −4), and −3(−2) = +6 (subtracting a negative). Choice A (0) would result from incorrectly computing (−2)² as −4, giving −4 + 6 + 2 = 4, not 0. Choice C (8) often comes from computing −3(−2) as −6 instead of +6.

A set is closed under subtraction if subtracting any two elements always produces a result within the same set. Integers are closed: integer − integer is always an integer. Natural numbers fail: 1 − 3 = −2, which is not a natural number. Whole numbers fail: 0 − 1 = −1, which is not a whole number. Positive even integers fail: 2 − 4 = −2, which is not positive.

Convert all values to decimals: −7/4 = −1.75, −0.5 = −0.5, 0 = 0, and 3/2 = 1.5. Plotting on a number line from left (least) to right (greatest): −1.75, −0.5, 0, 1.5. Choice B is wrong because −7/4 = −1.75 is less than −0.5, so −7/4 must appear first.

'The product of 6 and x' is 6x. 'Decreased by 4' means subtract 4 from that result, giving 6x − 4. Choice A, 6(x − 4), means 'six times the quantity x minus 4,' which equals 6x − 24 — a much larger subtraction. Choice C reverses the subtraction order, giving a different value.

Distribute each term: −3(2x − 1) = −6x + 3 and 4(x + 2) = 4x + 8. Combine like terms: (−6x + 4x) + (3 + 8) = −2x + 11. A frequent error is computing −3 × (−1) = −3 instead of +3, which would incorrectly give −2x + 5. Choice B results from adding the x coefficients (6 + 4) instead of subtracting.

Distribute the negative sign to every term inside the parentheses: −(x − 3) = −x + 3. Then add to 5: 5 + (−x + 3) = 8 − x. Choice A is the most common error — treating −(x − 3) as −x − 3, which gives 2 − x. The negative must be distributed to both x and −3, turning −3 into +3.

Substitute y = −4: 3|−4| − (−4). The absolute value |−4| = 4. So: 3(4) − (−4) = 12 + 4 = 16. Two important steps: absolute value makes −4 positive, and subtracting a negative number is the same as adding. Choice C (8) likely comes from ignoring the − y term and computing only 3|y| = 12 — 12 ≠ 8, so that error compounds further.

Work from the innermost parentheses outward. First expand b − (c − d) = b − c + d. Then subtract: a − (b − c + d) = a − b + c − d. The outer negative distributes to all three terms: −b becomes −b, −(−c) becomes +c, and −(+d) becomes −d. Choice B results from incorrectly distributing the outer negative to c as well.

When both x and y are negative, their product x · y = (negative)(negative) = positive, always. x + y = negative + negative = always negative. x − y can be either sign: if x = −1 and y = −3, then x − y = −1 − (−3) = 2 > 0, but if x = −3 and y = −1, then x − y = −2 < 0. For x² + y: x² is positive but y is negative, so the sum could go either way.

Work from the innermost grouping outward. Step 1: (3 − 1) = 2. Step 2: 2² = 4. Step 3: inside the brackets, 4 − 4 = 0. Step 4: 2[0] = 0. Step 5: 5 − 0 + 3 = 8. Choice B (2) results from evaluating (3 − 1²) = 2 instead of (3 − 1)² = 4, applying the exponent before subtracting inside the innermost group.

Distribute each part: 2(x + 3) = 2x + 6, and −(x − 1) = −x + 1. Combine like terms: 2x − x = x and 6 + 1 = 7, giving x + 7. Choice A (x + 5) is the most common error: distributing the negative as −(x − 1) = −x − 1 instead of −x + 1 — the negative must be applied to both terms, turning −1 into +1. Choice C (3x + 5) results from treating −(x − 1) as +(x − 1) = x − 1.

Distribute: 3x − 3a + 2x = 5x − 12. Combine x terms on the left: 5x − 3a = 5x − 12. Since 5x appears on both sides, subtract 5x from both sides: −3a = −12. Divide by −3: a = 4. Choice B (−4) results from dividing 12 by 3 without preserving both negative signs: −12 ÷ (−3) = +4, not −4.

Expand each expression: 2(x + 3) = 2x + 6. This matches 2x + 6 directly, and also equals 6 + 2x by the commutative property of addition. However, 2(x − 3) = 2x − 6, which differs from 2x + 6 by 12. The error is assuming that 2(x − 3) and 2(x + 3) are equivalent because they look similar — the sign inside the parentheses changes the constant term entirely.

Since p > 0 and q < 0, we know −q > 0. Therefore p − q = p + (−q) = positive + positive > 0, which is always true. Choice A is not always true: if p = 1 and q = −5, then p + q = −4 < 0. Choice B is never true: a positive times a negative is always negative. Choice D is never true: squares are non-negative, so p² + q² ≥ 0, never less than 0.

The commutative property states that the order of addends can be reversed without changing the sum: a + b = b + a. Choice A shows the associative property (regrouping), Choice C shows the additive identity property, and Choice D shows the additive inverse property.

Natural numbers are the positive counting numbers: 1, 2, 3, 4, and so on. The number 8 qualifies. The number −5 is negative, 0 is a whole number but not a natural number, and 2/3 is a fraction and not an integer.

Absolute value measures a number's distance from zero on the number line and is always non-negative. The distance from −15 to 0 is 15, so |−15| = 15. The result of an absolute value is never negative.

The associative property allows regrouping of factors without changing the product: a × (b × c) = (a × b) × c. The order of the three factors is unchanged on both sides — only the grouping (parentheses) changes. The commutative property would change the order of factors, not their grouping.

A constant term contains no variable. In 7x² − 3x + 5, the term 5 stands alone with no variable factor. The terms 7x² and −3x both contain the variable x. The number 7 is the coefficient of x², not a standalone constant term in this expression.

An irrational number cannot be written as a ratio of two integers; its decimal expansion neither terminates nor repeats. √5 ≈ 2.2360679... fits that description. In contrast, 0.75 = 3/4 (rational), −4 = −4/1 (rational), and √9 = 3 (rational).

Order of operations requires multiplication before addition. First compute 3 × 2 = 6, then add 4 to get 10. Choice B (14) results from incorrectly adding 4 + 3 = 7 first, then multiplying by 2 — a common order-of-operations error.

Terms in a polynomial are the parts separated by addition or subtraction. The expression 4x² − 7x + 3 has three terms: 4x², −7x, and 3. A common mistake is counting the number of distinct variables or the number of operations rather than the number of additive parts.

The distributive property states a(b + c) = ab + ac. Here, 3 is multiplied by each term inside the parentheses: 3 · x + 3 · 4 = 3x + 12. The commutative and associative properties change the order or grouping of existing operations, but they do not distribute multiplication across addition.

Substitute and evaluate: 2(3)² − (3)(−2) = 2(9) − (−6) = 18 + 6 = 24. A frequent error is treating −(3)(−2) as −6 instead of +6, yielding 18 − 6 = 12 (Choice A). Remember that subtracting a negative equals adding a positive.

Converting to decimals: −1/3 ≈ −0.333 and 1/4 = 0.25. The number 0 lies strictly between these two values. Choice A (−1/2 = −0.5) is less than −1/3, Choice B (1/3 ≈ 0.333) is greater than 1/4, and Choice D (−2/5 = −0.4) is also less than −1/3.

Group and combine like terms: (4x² − 2x²) + (3x + x) + (−5) = 2x² + 4x − 5. Choice B incorrectly adds 4 + 2 = 6 for the x² terms instead of subtracting. Choice C incorrectly combines the x terms as 3x − x = 2x, but both x terms carry positive signs and must be added.

The multiplicative inverse (reciprocal) of a number n is 1/n, chosen so that n × (1/n) = 1. The reciprocal of −3/4 is −4/3 because (−3/4)(−4/3) = 12/12 = 1. Choice A (3/4) is wrong because (−3/4)(3/4) = −9/16 ≠ 1. The sign of the original number must be preserved in its reciprocal.

Apply order of operations: exponents first (3² = 9), then multiplication and division left to right (4 × 2 = 8 and 6 ÷ 2 = 3), then addition and subtraction: 9 + 8 − 3 = 14. Choice B (10) comes from working left to right without respecting operation precedence. Choice C results from treating 3² as 3 × 2 = 6.

Every integer can be written as a fraction with denominator 1 (for example, −3 = −3/1), so all integers belong to the rational numbers. Natural numbers include only positive integers, and whole numbers include only non-negative integers — neither contains negative integers. Irrational numbers cannot be expressed as fractions.

The product of two odd numbers is always odd, so n² = n × n is odd whenever n is odd. The expression 2n is always even because multiplying any integer by 2 produces an even result. Adding 1 to an odd number (n + 1) always yields an even result. And n + n = 2n is always even for the same reason as 2n.

'Quotient of x and 3' means x ÷ 3. 'Eight more than' that quotient means 8 is added afterward: x ÷ 3 + 8. Choice B adds 8 to x before dividing, which changes the value. Choice D adds 8 to the divisor 3 before dividing, which is also a misreading of the phrase.

Distributing gives 12 − 2(x − 5) = 12 − 2x + 10 = 22 − 2x. Choices A, B, and C all simplify to 22 − 2x. Choice D equals 12 − 2x − 10 = 2 − 2x, which is not equal to 22 − 2x. The error in D is failing to distribute the negative sign correctly: −2 times −5 is +10, not −10.

Expand the left side: 4x + 4a − 2x = 2x + 4a. For the equation 2x + 4a = 2x + 12 to be true for all x, the x terms already match, so the constants must match: 4a = 12, giving a = 3. A common error is dividing 12 by 2 to get 6, confusing the coefficient 4 of a with 2.

Without parentheses, −2³ means −(2³) = −8, because the exponent applies only to 2, not to the negative sign. With parentheses, (−2)³ = −8 as well. So −8 + (−8) = −16. Choice B (0) is a common error from treating −2³ as (−2)³ = −8, then incorrectly negating it to +8. The parentheses make all the difference.

If r is rational and i is irrational, their sum r + i must be irrational. If it were rational, then (r + i) − r = i would be rational — a contradiction. The other choices are false: √2 + (−√2) = 0 is rational, which disproves both A and D. And √2 × √2 = 2 is rational, which disproves B.

Work from the innermost grouping outward. Inside the brackets: 3 − (2x − 4) = 3 − 2x + 4 = 7 − 2x. Apply the outer negative sign: −(7 − 2x) = −7 + 2x. Then add x: −7 + 2x + x = 3x − 7. A common error is distributing the outer negative to only the first term of the bracket, leaving the result incomplete.

When |A| = |B|, either A = B or A = −B. Setting x − 4 = x + 2 gives −4 = 2, which is impossible. Setting x − 4 = −(x + 2) gives x − 4 = −x − 2, so 2x = 2 and x = 1. Checking: |1 − 4| = |−3| = 3 and |1 + 2| = 3. Both sides equal 3, confirming x = 1.

Evaluate each part using order of operations: −|−6| = −6; (−3)² = 9 and 9 ÷ 3 = 3; 2³ = 8. Combine: −6 + 3 − 8 = −11. Choice B (1) results from treating −|−6| as +6. Choice C (−7) results from an order-of-operations error: adding −6 + 9 = 3 before dividing, then (3 ÷ 3) − 8 = −7.

From a > b and b > c, the transitive property gives a > c, so a − c > 0 is always true. Choice A fails when a = 5, b = 3, c = 1: both differences equal 2, so neither is strictly greater. Choice B fails when b or c is negative, reversing the direction of the inequality. Choice C fails when a = 10, b = 8, c = 1: a + c = 11 is less than 2b = 16.

The commutative property of addition states that changing the order of addends does not change the sum: a + b = b + a. Choice A shows the associative property (regrouping), choice C shows the additive identity property, and choice D shows the distributive property.

The coefficient is the numerical factor multiplied by the variable. In 7x, the coefficient of x is 7. The −3 is the constant term, not a coefficient of x. Coefficients can be negative, but in this expression the coefficient of x is positive 7.

√5 is irrational because 5 is not a perfect square, so its square root cannot be expressed as a ratio of two integers. In contrast, 0.333... = 1/3 (rational), √9 = 3 (rational), and −7/2 is already a ratio of integers (rational). A common mistake is to assume all square roots are irrational — only square roots of non-perfect squares are irrational.

Absolute value represents the distance from zero on the number line, which is always non-negative. |−15| = 15 because −15 is 15 units away from zero. Absolute value never produces a negative result, so choice A is incorrect.

A constant term has no variable part — its value does not change regardless of the variable's value. In 5x² − 3x + 8, the term 8 is the constant. The 5 and −3 are coefficients of variable terms, and x² is a variable factor, not a constant term.

The associative property of multiplication states that regrouping factors does not change the product: a × (b × c) = (a × b) × c. The numbers and their order stay the same — only the grouping (parentheses) changes. The commutative property would change the order of the factors, not the grouping.

Whole numbers include zero and all positive integers: {0, 1, 2, 3, ...}. Natural numbers typically start at 1 and do not include zero. Positive integers also exclude zero. Zero is a rational number (0 = 0/1), not irrational.

Substitute x = −2: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15. A common error is computing −2² instead of (−2)², which gives 2(−4) + 6 + 1 = −1 (choice B). Note that (−2)² = 4 because squaring makes the result positive.

The distributive property states that a(b + c) = ab + ac. Here, 5 is multiplied by each term inside the parentheses: 5 · x = 5x and 5 · 3 = 15. The commutative property changes order, the associative property changes grouping, and the identity property involves adding zero or multiplying by one.

Distribute each factor: 3(2x − 4) = 6x − 12 and 2(x + 5) = 2x + 10. Then combine: (6x + 2x) + (−12 + 10) = 8x − 2. Choice B results from a sign error (−12 + 10 = +2). Choice D results from subtracting instead of adding the constants (−12 − 10 = −22).

Following order of operations: parentheses first — (3 + 6) = 9. Then left to right for multiplication and division: 18 ÷ 9 = 2, then 2 × 2 = 4. Finally, subtraction: 4 − 1 = 3. Choice D (11) results from ignoring the parentheses and computing 18 ÷ 3 = 6 first, then 6 × 2 − 1 = 11.

'Three times a number n' is 3n. 'Five less than' that means subtracting 5 from it, giving 3n − 5. Choice A reverses the subtraction (5 − 3n), which means 'three times n less than five.' Choice C means 'three times the quantity n minus five,' which is a different expression.

√7 ≈ 2.646 and √10 ≈ 3.162. A number between them must be greater than 2.646 and less than 3.162. Only 2.8 satisfies this condition. Choice A (2.5) is less than √7, and choices C (3.2) and D (3.5) are both greater than √10.

The additive inverse property states that every real number added to its opposite (negative) equals zero: x + (−x) = 0. Choice A shows the commutative property of addition, choice C shows the additive identity property, and choice D shows the multiplicative inverse property.

Combine like terms: (2a − a) + (3b + 5b) = a + 8b. The coefficient of b is 8. A common error is choosing 3 or 5 (using only one of the b-coefficients) or 15 (multiplying 3 × 5 instead of adding).

Apply the distributive property: −2 × 3 = −6 and −2 × (−x) = +2x. So −2(3 − x) = −6 + 2x. Choice A results from treating −2(−x) as −2x (sign error). Choice C results from distributing the negative incorrectly on the constant only.

The degree of a polynomial is the highest exponent of the variable. The term 4x³ has degree 3, the term −7x has degree 1, and the constant 2 has degree 0. Therefore the degree of the polynomial is 3. Choice D (4) is the leading coefficient, not the degree.

(−3)² = (−3)(−3) = 9 and 3² = 9, so 9 − 9 = 0. A common error is computing −3² instead of (−3)²: without parentheses, −3² = −(3²) = −9. That error would give −9 − 9 = −18 (choice B). The parentheses in (−3)² are essential — they indicate the negative sign is part of the base being squared.

Expand (x + 3)² = x² + 6x + 9, then multiply by 2: 2x² + 12x + 18. Subtract 2x²: 2x² + 12x + 18 − 2x² = 12x + 18. Choice B (6x + 9) results from forgetting to multiply by 2 and canceling x² terms. Choice D (18) results from incorrectly expanding (x + 3)² as x² + 9, omitting the middle term 6x.

When a < 0 and b < 0: (A) a + b is the sum of two negatives, which is always negative — not positive. (B) ab is the product of two negatives, which is always positive — not negative. (C) a/b is a negative divided by a negative, which is always positive — this is always true. (D) a − b could be positive or negative depending on the specific values (e.g., −1 − (−2) = 1 > 0, but −2 − (−1) = −1 < 0).

Work from the innermost grouping outward. First simplify inside the brackets: 3x − (x + 5) = 3x − x − 5 = 2x − 5. Then: 4 − 2(2x − 5) + 2x = 4 − 4x + 10 + 2x = (−4x + 2x) + (4 + 10) = −2x + 14. Choice B results from distributing −2 as if the inner sign were positive: 3x − x + 5 = 2x + 5, then 4 − 2(2x + 5) + 2x = 4 − 4x − 10 + 2x = −2x − 6.

Choice C is always true: rational + irrational = irrational (the irrationality cannot be cancelled). Choice A is false: √2 + (−√2) = 0, which is rational. Choice B is false: √2 × √2 = 2, which is rational. Choice D is false: 0 × π = 0, which is rational — multiplying any irrational by zero yields zero.

Following order of operations — parentheses first: 2³ − 2 × 3 = 8 − 6 = 2. Then evaluate exponents outside parentheses: 2³ = 8 and 4² = 16. Now left to right for multiplication and division: 8 × 3 = 24 and 16 ÷ 2 = 8. Finally, subtraction: 24 − 8 = 16. Choice A (8) results from computing 24 − 16 = 8 before dividing. Choice D (24) results from stopping after the first multiplication.

Expand the left side: 3(2x − a) + a = 6x − 3a + a = 6x − 2a. Set equal to the right side: 6x − 2a = 6x − 8. Since this holds for all x, the x-terms match automatically; equate constants: −2a = −8, so a = 4. Choice B (−4) results from a sign error such as writing 2a = −8. Choice D (8) could result from errors combining the a-terms: incorrectly getting −3a + a = −a instead of −2a.

Test (1, 1): left side (1 + 1)² = 4; right side 1² + 1² = 2. Since 4 ≠ 2, this disproves the claim. The other pairs all have one value equal to zero: (a + 0)² = a² + 0², so those cases satisfy the equation and cannot be counterexamples. The correct expansion is (a + b)² = a² + 2ab + b², which equals a² + b² only when ab = 0.

The commutative property of addition states that a + b = b + a — the order of addends can be swapped without changing the sum. Choice C shows 7 + 3 = 3 + 7, which matches this pattern. Choice A is the associative property (regrouping), Choice B is the identity property (adding zero), and Choice D is the distributive property.

The coefficient is the numerical factor multiplied by the variable. In 9x², the coefficient is 9. The exponent is 2, which indicates the power of x but is not the coefficient. 18 would result from incorrectly multiplying 9 × 2.

√11 is irrational because 11 is not a perfect square and cannot be expressed as a ratio of two integers. -7 = -7/1 is rational, 0.25 = 1/4 is rational, and √9 = 3 = 3/1 is rational. A common mistake is treating √9 as irrational without evaluating it first.

Like terms must have exactly the same variable(s) raised to the same power(s). -6x and 9x both contain x to the first power, so they are like terms and can be combined. 3x and 3y differ in variable, 5x² and 5x differ in exponent, and 4xy and 4x differ in the number of variable factors.

The multiplicative identity property states that any number multiplied by 1 equals that number: a × 1 = a. Choice C shows 8 × 1 = 8, which matches. Choice A shows the zero property of multiplication, and Choice B shows the multiplicative inverse property.

Absolute value represents the distance from zero on the number line and is always non-negative. The absolute value of any negative number is its positive counterpart, so |-15| = 15. Distance cannot be negative, which is why the answer is never -15.

-6 is an integer (a negative whole number), which makes it also a rational number (expressible as -6/1) and a real number. It is not a natural number (which start at 1) nor a whole number (which start at 0), and it is not irrational because it can be written as a fraction.

A constant term has no variable factor — its value does not change. 11 is the only term without an x. 5x³ and -3x are variable terms. 5 is the coefficient of the term 5x³, not a separate standalone constant term in the expression.

Using order of operations (PEMDAS), perform multiplication and division before addition and subtraction: 3 × 4 = 12 and 6 ÷ 2 = 3. Then evaluate left to right: 2 + 12 − 3 = 11. Choice B (7) results from working strictly left to right: 2 + 3 = 5, then 5 × 4 = 20, then 20 − 6 = 14, then 14 ÷ 2 = 7 — incorrectly ignoring operation precedence.

The distributive property states a(b + c) = ab + ac. Here, 4 is multiplied by each term inside the parentheses: 4 × x = 4x and 4 × 7 = 28. The commutative property changes the order of factors, and the associative property changes their grouping — neither applies here.

Substitute x = -2: 3(-2)² − (-2) + 4 = 3(4) + 2 + 4 = 12 + 2 + 4 = 18. A common error is computing (-2)² = -4 (forgetting that squaring a negative gives a positive result), which would give 3(-4) + 2 + 4 = -6, matching Choice A.

'Product of n and 6' translates to 6n. 'Increased by twice the number' means adding 2n. Combined, the expression is 6n + 2n. Choice A misreads 'twice the number' as simply the constant 2. Choice D incorrectly factors out 6, changing the meaning of the phrase.

Distribute both terms: 5(2x − 3) = 10x − 15 and 3(x − 4) = 3x − 12. Then subtract: (10x − 15) − (3x − 12) = 10x − 15 − 3x + 12 = 7x − 3. Choice B (7x − 27) results from forgetting to distribute the negative sign, treating −3(x − 4) as −3x − 12 instead of −3x + 12.

3xy + 2x cannot be simplified because 3xy and 2x are not like terms — they have different variable factors (xy vs. x alone). Choice A simplifies to 7x, Choice B simplifies to a + 4, and Choice D simplifies to 3x². Only 3xy + 2x has no combinable like terms.

Identify and combine like terms: −3x² and −3x² are like terms, so −3x² + (−3x²) = −6x². The terms 7x and 2 have no other like terms to combine. The result is −6x² + 7x + 2. Choice B results from only counting one −3x² term instead of combining both.

The associative property of addition states (a + b) + c = a + (b + c) — the grouping of addends changes without affecting the order or sum. Here the parentheses shift around different terms. The commutative property would change the order of the terms themselves (e.g., 5 + x = x + 5), not the grouping.

Substitute a = 2 and b = -3: (2)² − (2)(−3) + (−3)² = 4 − (−6) + 9 = 4 + 6 + 9 = 19. A frequent error is treating −ab as −(2)(3) = −6 rather than −(2)(−3) = +6, which produces 4 − 6 + 9 = 7 (Choice A).

Substitute p = -1 and q = 2: p³ = (-1)³ = -1; 2p²q = 2(1)(2) = 4; pq² = (-1)(4) = -4; q³ = 8. Then: -1 − 4 + (-4) − 8 = -17. A common error is computing (-1)³ = 1 (forgetting that an odd power of a negative number is negative), which shifts the result to -15.

The error is in Step 1. Distributing -2 across (x − 4) gives -2x + 8 because -2 × (-4) = +8, not -8. The correct simplification is 3x − 2x + 8 + 5 = x + 13. Step 2 follows correctly from the flawed Step 1 (x − 8 + 5 = x − 3), so the sole error is the sign of the constant produced by distribution.

A product is positive when both factors are positive or both are negative — either way, the two numbers share the same sign. Choice A is only partially correct (it ignores the case where both are negative). Choice B is also only partially correct (it ignores both-positive). Choice D would produce a negative product, not a positive one.

Work from the innermost grouping outward. First, distribute the negative: -(4 - x) = -4 + x, so the bracket becomes 2x − 4 + x = 3x − 4. Multiply by 3: 3(3x − 4) = 9x − 12. Finally, subtract 5x: 9x − 12 − 5x = 4x − 12. Choice B results from incorrectly treating -(4 - x) as -4 - x, which gives 3[x - 4] - 5x = 3x - 12 - 5x = -2x - 12.

The closure property states that performing an operation on elements of a set always produces a result within that same set. Choice B expresses that the sum of any two real numbers is itself a real number — that is closure. Choice A is commutativity, Choice C is the additive inverse property, and Choice D is associativity.

x = 0 is the best counterexample because 0² = 0, which is not positive — it is zero. The student's claim should say 'non-negative' rather than 'positive.' Squaring −3, 1/2, or −10 all yield positive results (9, 1/4, and 100 respectively), so none of those disprove the claim.

The absolute value |a| equals a only when a ≥ 0. Setting 2x − 6 ≥ 0 and solving: 2x ≥ 6, so x ≥ 3. When x < 3, the expression 2x − 6 is negative, so |2x − 6| = -(2x − 6) = 6 − 2x, which does not equal 2x − 6. The boundary value x = 3 must be included because |0| = 0 = 2(3) − 6.

a² + b² ≥ 2ab is always true because (a − b)² ≥ 0 expands to a² − 2ab + b² ≥ 0, which rearranges to a² + b² ≥ 2ab. Choice A fails when a = b = 0 (gives 0 > 0, which is false). Choice B fails when a = b (gives |0| > 0, which is false). Choice D fails when a = b = 1 (gives 2 > 2, which is false).

√3 ≈ 1.732..., a non-terminating, non-repeating decimal, which makes it irrational. √25 = 5 (a rational integer), 7/8 = 0.875 (a terminating decimal, so rational), and −4 is an integer and therefore rational.

The coefficient is the numerical factor multiplied by the variable part. In 7y², the coefficient is 7. The 2 is the exponent, not the coefficient. The value −3 is the coefficient of y, and 5 is the constant term.

Following order of operations, perform division and multiplication before addition: 18 ÷ 3 = 6 and 4 × 2 = 8, then 6 + 8 = 14. A common error is working strictly left to right: (18 ÷ 3 + 4) × 2 = 10 × 2 = 20, which violates the correct order.

Natural numbers are {1, 2, 3, ...} and whole numbers are {0, 1, 2, 3, ...}. The only whole number excluded from the natural numbers is 0. The number 1 belongs to both sets. The value −1 is an integer but not a whole number. The value 1/2 is neither whole nor natural.

The additive inverse property states that every real number a has an opposite −a such that their sum equals zero: a + (−a) = 0. The additive identity states a + 0 = a (adding zero does not change the value). The multiplicative inverse is about multiplying by a reciprocal to get 1, not 0.

Like terms must have identical variable parts — the same variables raised to the same exponents. Both 3mn and −8mn contain the variable combination mn to the first power, so they are like terms. The pair 4x²y and 4xy² have different exponent arrangements. The pair 6a and 6b have different variables. The pair 2x³ and 2x² differ in the exponent on x.

Evaluate the exponent first: 2³ = 8. Then multiply: 8 × 3 = 24. Then subtract: 24 − 10 = 14. A common error is computing 2 × 3 first and then cubing: (6)³ − 10 = 216 − 10 = 206, which incorrectly ignores the exponent's priority over multiplication.

Distribute each factor: 4(3x + 2) = 12x + 8 and −2(x − 5) = −2x + 10. Combine like terms: 12x − 2x = 10x and 8 + 10 = 18, giving 10x + 18. A common error is writing −2(x − 5) = −2x − 10, forgetting that multiplying a negative by a negative yields a positive.

Substitute and evaluate: 2(3)² + (3)(−2) − (−2)² = 2(9) + (−6) − (4) = 18 − 6 − 4 = 8. Note that n² = (−2)² = 4, a positive value. A common error is treating −n² as −(−4) = 4, which would add rather than subtract 4.

Since √5 ≈ 2.236, it falls between 0 and 3. The correct order is: −3 < −1/2 < 0 < √5 ≈ 2.236 < 3. Choice B incorrectly places 3 before √5. Choices C and D misorder the negative numbers, since −3 < −1/2 (farther left on the number line means less).

The distributive property states a(b + c) = ab + ac. Here, 3(x + 4) = 3·x + 3·4 = 3x + 12. The commutative property switches order of terms (e.g., ab = ba), and the associative property changes grouping of terms — neither involves multiplying through parentheses.

The subtotal for 5 notebooks is 5n. A $3 discount applied to the total gives 5n − 3. Choice B, 5(n − 3), would mean the price per notebook is reduced by $3 before multiplying — that is a per-item discount, not a discount on the total.

Work inside the parentheses first: 3 − 5 = −2. Apply the exponent: (−2)² = 4. Multiply: −2 × 4 = −8. Add: −8 + 4 = −4. A common error is applying the exponent to −2(3 − 5) before squaring: [−2(−2)]² = 4² = 16, which violates order of operations.

−√16 = −4, a negative integer. As an integer, it is also rational (written as −4/1) and real. It is NOT whole (whole numbers start at 0 and include only non-negatives) and NOT natural (natural numbers are positive). It is NOT irrational because it simplifies exactly to −4.

'The product of four and x' is 4x. 'Seven less than' means subtract 7 from that product, giving 4x − 7. Choice A, 7 − 4x, subtracts the product from 7 — the reverse. Choice C, 4(x − 7), incorrectly subtracts 7 from x before multiplying by 4.

Convert 2x to (8/4)x so all terms share a common denominator: (3/4)x + (8/4)x − (1/4)x = (3 + 8 − 1)/4 · x = (10/4)x = (5/2)x. Simplifying 10/4 by dividing numerator and denominator by 2 gives 5/2. Choice A, 2x = (8/4)x, ignores the (3/4)x and −(1/4)x terms.

Distribute: 2(3x − 1) = 6x − 2. Then add 4: 6x − 2 + 4 = 6x + 2. Choice B, 6x − 6, results from incorrectly combining −1 and +4 as −5 before distributing. Choice C, 6x + 6, mistakenly distributes the 2 to the +4 outside the parentheses as well.

'The square of n' is n². 'Twice the square' is 2n². 'Three less than' means subtract 3 from that value: 2n² − 3. Choice A, 2n − 3, omits the squaring. Choice C, 3 − 2n², reverses the subtraction. Choice D, 2(n − 3)², incorrectly applies squaring after subtracting 3 from n.

Simplify inside the brackets first: 3x − (2 − 4x) = 3x − 2 + 4x = 7x − 2. Multiply by −2: −2(7x − 2) = −14x + 4. Expand the second term: 5(x + 1) = 5x + 5. Combine: −14x + 4 + 5x + 5 = −9x + 9. A common error is distributing only to 3x inside the brackets and missing −4x.

By standard convention, −3² = −(3²) = −(9) = −9 because the exponent applies only to the base 3, and the leading negative sign is separate. To square negative three as a unit, parentheses are required: (−3)² = 9. The student confused these two distinct expressions.

x² = (−2)² = 4, so x²y = 4 × 3 = 12. For the middle term: −2xz = −2(−2)(−1) = −2(2) = −4. For the last term: z³ = (−1)³ = −1. Total: 12 + (−4) + (−1) = 7. A common error is computing −2(−2)(−1) as positive 4 by misapplying the sign rules for multiplying three negatives.

Both √2 and −√2 are irrational numbers, yet their sum is 0 — a rational integer. This single counterexample is enough to disprove the claim. Choices A, C, and D all produce irrational sums, which are consistent with the claim but do not disprove it; only a counterexample can do that.

Distribute the negative sign to every term inside the parentheses: −(x) = −x, −(−y) = +y, and −(3) = −3. The result is −x + y − 3. Choice B incorrectly applies −(−y) = −y. Choice C incorrectly yields +3 for the last term. Choice D does not apply the negative sign at all.

Start with the innermost operation: (−1)³ = −1. Multiply: −1 × 4 = −4. Evaluate inside the brackets: 2 + (−4) = −2. Multiply by 3: 3 × (−2) = −6. Finally: 4 − (−6) = 4 + 6 = 10. A common error is treating subtraction as part of the multiplication before completing the bracket, giving 4 − 3 = 1 first, which violates order of operations.

Irrational numbers such as π and √2 are real numbers but are not rational, so it is false that every real number is rational. The other three statements are true: integers can be expressed as fractions (e.g., −4 = −4/1), natural numbers (1, 2, 3, ...) are a subset of whole numbers (0, 1, 2, ...), and the real number set includes both rational and irrational numbers.

The Commutative Property of Addition states that changing the order of addends does not change the sum: a + b = b + a. Here, 7 and 3 swap positions. This is different from the Associative Property (choice A), which involves changing the grouping of three or more numbers — for example, (7 + 3) + 2 = 7 + (3 + 2) — not the order.

Absolute value represents a number's distance from zero on the number line, which is always non-negative. Since −15 is 15 units away from zero, |−15| = 15. Choice A is a common error of simply removing the absolute value bars without considering that the result must be non-negative. Choice C is 15 squared, which is unrelated to the absolute value operation.

The Multiplication Property of Zero states that any number multiplied by zero equals zero: a × 0 = 0. Choice A illustrates the Multiplicative Identity Property (a × 1 = a), and choice B illustrates the Additive Identity Property (a + 0 = a). Choice D involves division, not multiplication, and is a separate property.

The Associative Property of Addition states that regrouping addends does not change the sum: (a + b) + c = a + (b + c). Here, the numbers 5, 2, and 8 appear in the same left-to-right order on both sides — only the parentheses move. The Commutative Property (choice A) would require the numbers themselves to change order, such as 5 + 2 = 2 + 5.

Combine like terms separately by grouping terms with the same variable. The m-terms: 5m − 2m = 3m. The n-terms: 3n + 4n = 7n. The simplified expression is 3m + 7n. Choice C incorrectly adds 5m + 2m = 7m instead of subtracting. Choice A incorrectly multiplies the variable parts together, treating unlike terms as if they can be combined into a single product.

Perimeter of a rectangle = 2l + 2w = 2(3x + 2) + 2(5) = 6x + 4 + 10 = 6x + 14. Choice C is a common error from computing only 2(length) = 6x + 4 and forgetting to add 2(width). Choice A comes from multiplying length times width (area), not adding twice each dimension. Choice D finds l + w without multiplying by 2.

The Multiplicative Inverse Property states that every non-zero number multiplied by its reciprocal equals 1: a × (1/a) = 1. Choice A shows the Multiplicative Identity Property (multiplying by 1 gives the number itself). Choice B shows the Multiplication Property of Zero. Choice D shows the Additive Inverse Property — it involves addition and zero, not multiplication and one.

Substitute x = −3: |3(−3) − 1| + 2(−3)² = |−9 − 1| + 2(9) = |−10| + 18 = 10 + 18 = 28. Choice B (10) results from computing only the absolute value term and ignoring 2x². Choice D (46) comes from treating 2x² as (2x)², giving (−6)² = 36, then 10 + 36 = 46. Choice A (−8) results from incorrectly treating (−3)² as −9, giving 2(−9) = −18, then 10 + (−18) = −8.

Choice D distributes to 3(2x) + 3(4) = 6x + 12, which does not equal 6x + 9. The other choices all equal 6x + 9: Choice A factors correctly as 3(2x + 3) = 6x + 9; Choice B expands to 6x + 8 + 1 = 6x + 9; Choice C applies the distributive property step by step to get 6x + 9. A common error is assuming any factored form with a 3 out front is equivalent without checking the constant inside.

Work from the innermost grouping outward. Innermost: 4 − (x + 1) = 4 − x − 1 = 3 − x. Next: 2x − (3 − x) = 2x − 3 + x = 3x − 3. Finally: 5 − 3(3x − 3) = 5 − 9x + 9 = 14 − 9x. Choice B (20 − 9x) results from incorrectly distributing: 4 − (x + 1) = 5 − x (adding instead of subtracting 1), leading to 5 − 3(3x − 5) = 20 − 9x. Choice C (−4 − 9x) results from a sign error in the last step: treating −3 × (−3) as −9 instead of +9, giving 5 − 9x − 9 = −4 − 9x.