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Praxis Math 5165 study materials
Praxis Math 5165 study resources

Praxis Math 5165 practice questions

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Question 1Number and Quantity, Algebra, and Functions

Evaluate 12 − 3(4 − 7) + 5.

  1. 14
  2. 26 Correct answer.
  3. −4
  4. 2

Explanation

Work inside the parentheses first: 4 − 7 = −3. Then multiply before adding or subtracting: −3(−3) = 9, so the expression becomes 12 + 9 + 5 = 26.

Question 2Number and Quantity, Algebra, and Functions

Compute 2/3 + 3/4 − 1/6.

  1. 1
  2. 4/5
  3. 5/4 Correct answer.
  4. 7/6

Explanation

Rewrite each fraction with the least common denominator 12: 8/12 + 9/12 − 2/12. Combining the numerators gives 15/12, which simplifies to 5/4.

Question 3Number and Quantity, Algebra, and Functions

Simplify (2x²y⁻¹)³.

  1. 8x⁶y³
  2. 8x⁵/y³
  3. 6x⁶/y³
  4. 8x⁶/y³ Correct answer.

Explanation

Raise each factor to the third power: 2³ = 8, (x²)³ = x⁶, and (y⁻¹)³ = y⁻³. Rewriting the negative exponent gives 8x⁶/y³.

Question 4Number and Quantity, Algebra, and Functions

What is (3 × 10⁵)(4 × 10⁻⁸) in scientific notation?

  1. 7 × 10⁻³
  2. 1.2 × 10⁻² Correct answer.
  3. 1.2 × 10³
  4. 1.2 × 10⁻³

Explanation

Multiply the coefficients and add the exponents: 3 · 4 = 12 and 10⁵ · 10⁻⁸ = 10⁻³. Since 12 × 10⁻³ is not in scientific notation, rewrite it as 1.2 × 10⁻².

Question 5Number and Quantity, Algebra, and Functions

Simplify √50 − √18 + √8.

  1. 4√10
  2. 2√2
  3. 6√2
  4. 4√2 Correct answer.

Explanation

Factor out perfect squares: √50 = 5√2, √18 = 3√2, and √8 = 2√2. Combining like radicals gives 5√2 − 3√2 + 2√2 = 4√2.

Question 6Number and Quantity, Algebra, and Functions

Multiply (3 + 2i)(4 − i).

  1. 14 + 5i Correct answer.
  2. 12 + 5i
  3. 10 + 5i
  4. 14 + 11i

Explanation

Expand using the distributive property: 12 − 3i + 8i − 2i². Because i² = −1, the term −2i² becomes +2, giving 14 + 5i.

Question 7Number and Quantity, Algebra, and Functions

Rationalize the denominator: 6/(√5 − √2).

  1. 6√5 + 6√2
  2. 2√5 − 2√2
  3. 2√5 + 2√2 Correct answer.
  4. 6√5 − 6√2

Explanation

Multiply numerator and denominator by the conjugate √5 + √2. The denominator becomes 5 − 2 = 3, so the expression is 6(√5 + √2)/3 = 2√5 + 2√2.

Question 8Number and Quantity, Algebra, and Functions

Two lights blink every 18 seconds and every 24 seconds. If they blink together now, in how many seconds will they next blink together?

  1. 432 min
  2. 6 min
  3. 72 min Correct answer.
  4. 36 min

Explanation

The lights coincide at common multiples of 18 and 24, and the next coincidence occurs at the least common multiple. Since 18 = 2 · 3² and 24 = 2³ · 3, the LCM is 2³ · 3² = 72.

Question 9Number and Quantity, Algebra, and Functions

A jacket costs $80. Its price is raised by 25%, and the new price is later reduced by 20%. What is the final price?

  1. $84
  2. $80 Correct answer.
  3. $100
  4. $76

Explanation

Successive percent changes multiply: 80(1.25) = 100, then 100(0.80) = 80. Because the 20% decrease is taken on the larger amount, the final price returns to the original $80.

Question 10Number and Quantity, Algebra, and Functions

A 64-liter mixture contains two chemicals in a ratio of 5 to 3. How many liters of the larger component are in the mixture?

  1. 44 L
  2. 40 L Correct answer.
  3. 24 L
  4. 32 L

Explanation

The ratio creates 5 + 3 = 8 equal parts, so each part is 64/8 = 8 liters. The larger component takes 5 parts: 5 · 8 = 40 liters.

Question 11Number and Quantity, Algebra, and Functions

Solve for x: 5(x − 3) = 2x + 6.

  1. 1
  2. −3
  3. 3
  4. 7 Correct answer.

Explanation

Distribute to get 5x − 15 = 2x + 6. Subtracting 2x from both sides gives 3x − 15 = 6, and adding 15 gives 3x = 21, so x = 7.

Question 12Number and Quantity, Algebra, and Functions

Solve the system 2x + 3y = 12 and x − y = 1.

  1. (3, 2) Correct answer.
  2. (1, 0)
  3. (2, 3)
  4. (4, 1)

Explanation

From the second equation x = y + 1. Substituting gives 2(y + 1) + 3y = 12, so 5y = 10 and y = 2, which makes x = 3.

Question 13Number and Quantity, Algebra, and Functions

What are the solutions of x² − 5x − 24 = 0?

  1. x = 12 and x = −2
  2. x = 6 and x = −4
  3. x = −8 and x = 3
  4. x = 8 and x = −3 Correct answer.

Explanation

Look for two numbers whose product is −24 and whose sum is −5; these are −8 and 3. Factoring gives (x − 8)(x + 3) = 0, so x = 8 or x = −3.

Question 14Number and Quantity, Algebra, and Functions

What does the discriminant of 2x² − 4x + 5 = 0 reveal about its solutions?

  1. two distinct real solutions
  2. two nonreal solutions Correct answer.
  3. exactly one real solution
  4. no real or complex solution

Explanation

The discriminant is b² − 4ac = 16 − 40 = −24. A negative discriminant means the two solutions are complex conjugates, not real numbers.

Question 15Number and Quantity, Algebra, and Functions

Solve |2x − 5| = 9.

  1. x = 7 or x = −2 Correct answer.
  2. x = −2 only
  3. x = 7 or x = 2
  4. x = 7 only

Explanation

An absolute value equation splits into two cases: 2x − 5 = 9 gives x = 7, and 2x − 5 = −9 gives x = −2. Both values check in the original equation.

Question 16Number and Quantity, Algebra, and Functions

Solve the inequality −3x + 4 > 19.

  1. x > 5
  2. x < −5 Correct answer.
  3. x > −5
  4. x < 5

Explanation

Subtract 4 from both sides to get −3x > 15. Dividing by −3 reverses the inequality symbol, so x < −5.

Question 17Number and Quantity, Algebra, and Functions

If f(x) = 2x² − 3x + 1, what is f(−2)?

  1. −1
  2. 11
  3. 3
  4. 15 Correct answer.

Explanation

Substitute −2 for every x: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1. This totals 15.

Question 18Number and Quantity, Algebra, and Functions

What is the domain of f(x) = √(3x − 12)?

  1. x ≤ 4
  2. x ≥ 12
  3. x ≥ 4 Correct answer.
  4. all real numbers

Explanation

The radicand of a square root must be nonnegative, so 3x − 12 ≥ 0. Solving gives x ≥ 4.

Question 19Number and Quantity, Algebra, and Functions

If f(x) = x² + 1 and g(x) = 2x − 3, what is g(f(x))?

  1. x² − 2
  2. 2x² + 2
  3. 2x² − 1 Correct answer.
  4. 4x² − 12x + 10

Explanation

Compose from the inside out: g(f(x)) = 2(x² + 1) − 3. Simplifying gives 2x² + 2 − 3 = 2x² − 1.

Question 20Number and Quantity, Algebra, and Functions

If f(x) = (2x − 1)/(x + 3), which formula gives f⁻¹(x)?

  1. (2x + 1)/(x − 3)
  2. (x + 3)/(2x − 1)
  3. (3x − 1)/(x + 2)
  4. (3x + 1)/(2 − x) Correct answer.

Explanation

Write y = (2x − 1)/(x + 3) and solve for x: y(x + 3) = 2x − 1, so x(y − 2) = −1 − 3y. Dividing gives x = (3y + 1)/(2 − y), and swapping y for x yields the inverse.

Question 21Number and Quantity, Algebra, and Functions

What is the slope of the line passing through (2, −1) and (6, 7)?

  1. 1/2
  2. −2
  3. 3/2
  4. 2 Correct answer.

Explanation

Slope is the change in y divided by the change in x: (7 − (−1))/(6 − 2) = 8/4 = 2.

Question 22Number and Quantity, Algebra, and Functions

A bacterial culture starts with 800 cells and triples every 5 hours. Which expression gives the number of cells after t hours?

  1. 800 · 5^(t/3)
  2. 3 · 800^(t/5)
  3. 800 · 3^(5t)
  4. 800 · 3^(t/5) Correct answer.

Explanation

In an exponential model with base b and period h, the count is P · b^(t/h). Here the base is 3 and the period is 5 hours, giving 800 · 3^(t/5).

Question 23Number and Quantity, Algebra, and Functions

If log_b 2 = 0.43 and log_b 5 = 1.00, what is log_b 20?

  1. 1.43
  2. 1.86 Correct answer.
  3. 0.86
  4. 2.00

Explanation

Since 20 = 4 · 5, log_b 20 = log_b 4 + log_b 5. Because log_b 4 = 2 log_b 2 = 0.86, the result is 0.86 + 1.00 = 1.86.

Question 24Number and Quantity, Algebra, and Functions

The polynomial p(x) = 2x³ − 5x² + ax − 7 has remainder 5 when divided by x − 3. What is a?

  1. a = −1
  2. a = 1 Correct answer.
  3. a = 5
  4. a = 3

Explanation

By the remainder theorem, the remainder equals p(3): 54 − 45 + 3a − 7 = 3a + 2. Setting 3a + 2 = 5 gives a = 1.

Question 25Number and Quantity, Algebra, and Functions

Simplify (x² − 9)/(x² + x − 12), assuming the expression is defined.

  1. (x + 3)/(x − 4)
  2. (x − 3)/(x + 4)
  3. (x − 3)/(x − 4)
  4. (x + 3)/(x + 4) Correct answer.

Explanation

Factor both parts: the numerator is (x − 3)(x + 3) and the denominator is (x + 4)(x − 3). Canceling the common factor (x − 3) leaves (x + 3)/(x + 4).

Question 26Number and Quantity, Algebra, and Functions

Simplify (3x²y⁻¹)³.

  1. 27x⁶/y³ Correct answer.
  2. 27x⁵/y³
  3. 9x⁶/y³
  4. 27x⁶y³

Explanation

Cube each factor separately: 3³ = 27, (x²)³ = x⁶, and (y⁻¹)³ = y⁻³. Writing the negative exponent as a denominator gives 27x⁶/y³.

Question 27Number and Quantity, Algebra, and Functions

What is the value of i²⁷, where i is the imaginary unit?

  1. −1
  2. 1
  3. −i Correct answer.
  4. i

Explanation

Powers of i repeat in the cycle i, −1, −i, 1 with period 4. Since 27 divided by 4 leaves remainder 3, i²⁷ equals i³ = −i.

Question 28Number and Quantity, Algebra, and Functions

Which of the following numbers is irrational?

  1. 22/7
  2. √49
  3. √11 Correct answer.
  4. 0.333...

Explanation

An irrational number cannot be written as a ratio of two integers. Since 11 is not a perfect square, √11 has a nonterminating, nonrepeating decimal expansion.

Question 29Number and Quantity, Algebra, and Functions

A quantity increases from 40 to 50. What is the percent increase?

  1. 25% Correct answer.
  2. 20%
  3. 10%
  4. 80%

Explanation

Percent increase equals the change divided by the original amount, multiplied by 100. The change is 10 and the original amount is 40, so (10/40)(100%) = 25%.

Question 30Number and Quantity, Algebra, and Functions

Solve for x: 4(x − 2) = 3x + 5.

  1. 13 Correct answer.
  2. 3
  3. −13
  4. 1

Explanation

Distribute to get 4x − 8 = 3x + 5. Subtract 3x from both sides to get x − 8 = 5, then add 8 to obtain x = 13.

Question 31Number and Quantity, Algebra, and Functions

What is the slope of the line given by 6x − 3y = 12?

  1. 6
  2. −2
  3. 1/2
  4. 2 Correct answer.

Explanation

Solve the equation for y: −3y = −6x + 12, so y = 2x − 4. In slope-intercept form y = mx + b, the slope m is 2.

Question 32Number and Quantity, Algebra, and Functions

If f(x) = 3x − 5, what is f(4) + f(1)?

  1. 5 Correct answer.
  2. 2
  3. 7
  4. 3

Explanation

First find f(4) = 3(4) − 5 = 7. Then f(1) = 3(1) − 5 = −2, and 7 + (−2) = 5.

Question 33Number and Quantity, Algebra, and Functions

What is log₃ 81?

  1. 9
  2. 3
  3. 4 Correct answer.
  4. 27

Explanation

The logarithm log₃ 81 asks for the exponent to which 3 must be raised to obtain 81. Since 3⁴ = 81, the value is 4.

Question 34Number and Quantity, Algebra, and Functions

Simplify (x² − 4)/(x² + 5x + 6) · (x + 3)/(x − 2), assuming all denominators are nonzero.

  1. (x − 2)/(x + 3)
  2. x + 3
  3. (x + 2)/(x − 2)
  4. 1 Correct answer.

Explanation

Factor the quadratics: x² − 4 = (x − 2)(x + 2) and x² + 5x + 6 = (x + 2)(x + 3). After substituting, the factors (x − 2), (x + 2), and (x + 3) all cancel, leaving 1.

Question 35Number and Quantity, Algebra, and Functions

What is the vertex of the parabola y = x² − 8x + 13?

  1. (8, 13)
  2. (4, 3)
  3. (4, −3) Correct answer.
  4. (−4, −3)

Explanation

For y = ax² + bx + c, the vertex x-coordinate is −b/(2a). Here x = 8/2 = 4, and substituting gives y = 16 − 32 + 13 = −3, so the vertex is (4, −3).

Question 36Number and Quantity, Algebra, and Functions

Which statement describes the system 3x + 2y = 6 and 6x + 4y = 10?

  1. It has exactly two solutions.
  2. It has infinitely many solutions.
  3. It has exactly one solution.
  4. It has no solution. Correct answer.

Explanation

The second equation has coefficients that are twice those of the first equation, but its constant 10 is not twice 6. The lines are parallel and distinct, so the system is inconsistent.

Question 37Number and Quantity, Algebra, and Functions

Solve the inequality |x − 3| ≤ 7.

  1. −10 ≤ x ≤ 4
  2. x ≤ −4 or x ≥ 10
  3. x ≤ 10
  4. −4 ≤ x ≤ 10 Correct answer.

Explanation

The inequality |x − 3| ≤ 7 is equivalent to −7 ≤ x − 3 ≤ 7. Adding 3 to all parts gives −4 ≤ x ≤ 10.

Question 38Number and Quantity, Algebra, and Functions

If f(x) = x³ + 2, which formula gives f⁻¹(x)?

  1. x³ − 2
  2. ∛(x + 2)
  3. ∛x − 2
  4. ∛(x − 2) Correct answer.

Explanation

Set y = x³ + 2 and solve for x. Subtracting 2 gives x³ = y − 2, so x = ∛(y − 2), and swapping variables gives f⁻¹(x) = ∛(x − 2).

Question 39Number and Quantity, Algebra, and Functions

What is the domain of f(x) = √(x² − 4)?

  1. All real numbers
  2. −2 ≤ x ≤ 2
  3. x ≥ 2
  4. x ≤ −2 or x ≥ 2 Correct answer.

Explanation

The expression under the square root must be nonnegative, so x² − 4 ≥ 0. Factoring gives (x − 2)(x + 2) ≥ 0, which is true when x ≤ −2 or x ≥ 2.

Question 40Number and Quantity, Algebra, and Functions

A radioactive sample has mass 200 mg and decays by 12% each year. Which expression gives the mass after t years?

  1. 200(1.12)^t
  2. 200(0.88t)
  3. 200(0.12)^t
  4. 200(0.88)^t Correct answer.

Explanation

Decaying by 12% leaves 88% of the mass each year, so the yearly multiplier is 0.88. Exponential decay has the form initial value times (multiplier)^t.

Question 41Number and Quantity, Algebra, and Functions

Solve log₂(x − 3) = 4.

  1. 16
  2. 7
  3. 11
  4. 19 Correct answer.

Explanation

Rewrite the logarithmic equation in exponential form: x − 3 = 2⁴ = 16. Adding 3 gives x = 19.

Question 42Number and Quantity, Algebra, and Functions

Which of the following is a factor of x³ − 2x² − 3x + 10?

  1. x − 2
  2. x − 1
  3. x + 2 Correct answer.
  4. x + 1

Explanation

By the factor theorem, x − c is a factor when the polynomial evaluates to 0 at x = c. For x + 2, evaluate at x = −2: −8 − 8 + 6 + 10 = 0, so x + 2 is a factor.

Question 43Number and Quantity, Algebra, and Functions

Simplify (4 + 2i)/(1 − i).

  1. 3 + 3i
  2. 2 + i
  3. 3 + i
  4. 1 + 3i Correct answer.

Explanation

Multiply numerator and denominator by the conjugate 1 + i. The denominator becomes 1² + 1² = 2, and the numerator becomes 4 + 4i + 2i + 2i² = 2 + 6i, so the quotient is 1 + 3i.

Question 44Number and Quantity, Algebra, and Functions

Simplify √50 + √18.

  1. 68
  2. 8√2 Correct answer.
  3. 2√17
  4. 4√2

Explanation

Factor out perfect squares: √50 = √(25·2) = 5√2 and √18 = √(9·2) = 3√2. Adding like radicals gives 5√2 + 3√2 = 8√2.

Question 45Number and Quantity, Algebra, and Functions

What is the domain of f(x) = (x + 1)/(x² − 9)?

  1. x ≠ −1
  2. x ≠ 3 and x ≠ −3 Correct answer.
  3. x ≠ 9
  4. x ≠ 3 only

Explanation

A rational function is undefined where its denominator is zero. Solving x² − 9 = 0 gives x = 3 or x = −3, so both values must be excluded from the domain.

Question 46Number and Quantity, Algebra, and Functions

Solve 3^(2x) − 10·3^x + 9 = 0 for real x.

  1. x = 0 only
  2. x = 1 or x = 3
  3. x = 2 only
  4. x = 0 or x = 2 Correct answer.

Explanation

Let u = 3^x. Then u² − 10u + 9 = 0 factors as (u − 1)(u − 9) = 0, so u = 1 or u = 9. Since 3⁰ = 1 and 3² = 9, the solutions are x = 0 and x = 2.

Question 47Number and Quantity, Algebra, and Functions

If f(x) = (x + 2)/(x − 5), which formula gives f⁻¹(x)?

  1. (5x + 2)/(x − 1) Correct answer.
  2. (x − 2)/(x + 5)
  3. (5x − 2)/(x + 1)
  4. (x + 5)/(x − 2)

Explanation

Set y = (x + 2)/(x − 5) and solve for x. Cross-multiplying gives xy − 5y = x + 2, so x(y − 1) = 5y + 2 and x = (5y + 2)/(y − 1). Swapping variables gives f⁻¹(x) = (5x + 2)/(x − 1).

Question 48Number and Quantity, Algebra, and Functions

If x − 2 is a factor of x³ + kx² − 4x + 4, what is k?

  1. 1
  2. −1 Correct answer.
  3. −2
  4. 2

Explanation

By the factor theorem, substituting x = 2 must give 0. Thus 8 + 4k − 8 + 4 = 0, which simplifies to 4k + 4 = 0, so k = −1.

Question 49Number and Quantity, Algebra, and Functions

What is the remainder when p(x) = x⁴ − 3x³ + 2x − 5 is divided by x + 2?

  1. 15
  2. −15
  3. 31 Correct answer.
  4. −31

Explanation

The remainder theorem says the remainder is p(−2). Evaluating gives (−2)⁴ − 3(−2)³ + 2(−2) − 5 = 16 + 24 − 4 − 5 = 31.

Question 50Number and Quantity, Algebra, and Functions

Solve (x − 1)/(x + 2) ≥ 0.

  1. x > −2 or x ≤ 1
  2. −2 < x ≤ 1
  3. x < −2 or x ≥ 1 Correct answer.
  4. x ≤ −2 or x ≥ 1

Explanation

The critical values are x = 1, where the numerator is zero, and x = −2, where the denominator is zero and the expression is undefined. A sign chart shows the expression is positive for x < −2 and for x > 1, and it equals zero at x = 1, giving x < −2 or x ≥ 1.