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Quizzes > High School Quizzes > Mathematics

Absolute Value Practice Quiz Questions

Sharpen your skills with challenging practice problems

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art promoting The Absolute Value Adventure quiz, a high school math tool for test preparation.

This absolute value quiz helps you practice high school math and see what you know. Answer 20 quick questions on distance from zero, expressions, and simple equations with absolute value, then see which topics to review so you can spot gaps before a test and build confidence.

What does the absolute value of a number represent?
Its distance from zero on the number line
Its squared value
Its negative value
Its reciprocal
Absolute value measures the distance of a number from zero and is always non-negative. It does not alter the number's magnitude based on its sign.
What is the value of |5|?
-5
0
10
5
For any positive number, the absolute value is the number itself. Thus, |5| equals 5.
What is the value of |-7|?
-7
0
14
7
Absolute value converts negative numbers to their positive counterparts because it represents distance from zero. Hence, |-7| is 7.
What is the value of |0|?
Undefined
1
-1
0
The absolute value of 0 is 0 because its distance from itself is zero. This is true regardless of the function's properties.
Which of the following statements is always true about absolute values?
An absolute value represents the opposite of a number.
An absolute value is always non-negative.
An absolute value can be negative if the original number is negative.
The absolute value of a number is always equal to the number.
By definition, absolute values denote the distance from zero and are always non-negative. Even when the number is negative, its absolute value is positive.
Solve for x: |x| = 4.
x = 0
x = 4 or x = -4
x = 4
x = -4
The equation |x| = 4 indicates that x can be either 4 or -4 because both yield an absolute value of 4. This covers both the positive and negative scenarios.
Solve for x: |x - 2| = 5.
x = 2
x = -3
x = 7 or x = -3
x = 7
Solve |x - 2| = 5 by considering the two cases: x - 2 = 5 and x - 2 = -5, which result in x = 7 and x = -3 respectively. Both solutions satisfy the equation.
Solve for x: |3x + 1| = 10.
x = 3
x = 10/3
x = 3 or x = -11/3
x = -11/3
Break the equation into two cases: 3x + 1 = 10 and 3x + 1 = -10. These yield x = 3 and x = -11/3 respectively, both of which solve the original equation.
What is the simplified form of | - (x - 3)|?
-|x - 3|
|x - 3|
x - 3
-x + 3
Multiplying by -1 inside an absolute value does not change the result, so | - (x - 3)| simplifies directly to |x - 3|. This property is a basic rule of absolute values.
Simplify the expression | - 7| + |3|.
10
4
21
-4
Evaluating each absolute value separately gives | - 7| = 7 and |3| = 3. Their sum, 7 + 3, equals 10.
Solve: |2x - 6| = 0.
No solution
3
6
0
The absolute value of an expression equals zero only when the expression itself equals zero. Setting 2x - 6 = 0 gives x = 3.
Find the value of x if |x + 5| = 8.
x = 8
x = 3
x = -13
x = 3 or x = -13
Solve the equation by setting up x + 5 = 8 and x + 5 = -8, which yield x = 3 and x = -13 respectively. Both solutions satisfy the equation.
Evaluate |2 - 5| + |5 - 2|.
3
9
6
0
Calculate the absolute values separately: |2 - 5| equals 3 and |5 - 2| equals 3. Their sum is 3 + 3, resulting in 6.
Which equation represents the fact that the distance between x and 4 is 7?
|x + 4| = 7
|x - 4| = 7
x - 4 = 7
x = 4 ± 3
The equation |x - 4| = 7 directly states that the distance between x and 4 is 7. Absolute value is the correct tool for representing distance on the number line.
Solve for x: |x - 3| = |x + 1|.
No solution
3
-1
1
By considering the equation |x - 3| = |x + 1|, we deduce that setting x - 3 = -(x + 1) leads to 2x = 2 and hence x = 1. The other possibility does not yield a valid solution.
Solve the inequality |x - 3| < 4.
x > 7 or x < -1
-1 < x < 7
x = -1 or x = 7
-1 ≤ x ≤ 7
The inequality |x - 3| < 4 means that x is less than 4 units away from 3, resulting in the interval (-1, 7). The endpoints are not included because the inequality is strict.
Solve the inequality |2x + 1| ≥ 7.
-4 < x < 3
x < 3
x ≤ -4 or x ≥ 3
x > -4
The inequality |2x + 1| ≥ 7 splits into two cases: 2x + 1 ≥ 7 and 2x + 1 ≤ -7. Solving these yields x ≥ 3 and x ≤ -4 respectively.
Which of the following represents the piecewise definition of |x - 5|?
x + 5 if x ≥ 5, and -x - 5 if x < 5
x - 5 if x ≥ 5, and 5 - x if x < 5
5 - x if x ≥ 5, and x - 5 if x < 5
5 + x if x ≥ 5, and 5 - x if x < 5
The absolute value function splits based on the sign of its argument. For |x - 5|, the expression equals x - 5 when x is at least 5, and 5 - x when x is less than 5.
Solve for x: |x - 4| + |x + 2| = 10.
x = -4
x = 6 or x = -4
x = 6
No solution
By analyzing the equation over intervals defined by the critical points x = -2 and x = 4, the solutions found are x = 6 and x = -4. Both satisfy the original equation.
If |x - 1| = |1 - x|, which of the following is correct?
The equality holds only for x = 1
The equality holds only for x > 1
The equality holds for all real numbers x
The equality holds only for x < 1
The absolute value function is symmetric, meaning that |x - 1| is always equal to |1 - x| for any real number x. This symmetry is a fundamental property of absolute values.
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Study Outcomes

  1. Interpret absolute value expressions in mathematical problems.
  2. Simplify and evaluate expressions involving absolute values.
  3. Solve equations and inequalities that feature absolute values.
  4. Analyze problem contexts to determine the impact of absolute values on solutions.
  5. Apply algebraic techniques to multi-step absolute value problems.

Absolute Value Questions Review Cheat Sheet

  1. Absolute Value Basics - Think of absolute value as a "distance meter" on the number line: it tells you how far a number is from zero, and it never goes below zero. So whether you plug in |5| or |-5|, you'll always get 5 - pretty neat, right? Absolute Value in Algebra
  2. Key Properties - Absolute values follow a few golden rules: they're always non-negative (|a| ≥ 0), they're symmetric (|a| = | - a|), and they obey the triangle inequality (|a + b| ≤ |a| + |b|). These properties make absolute values a solid foundation for advanced proofs and problem solving. Properties of Absolute Value
  3. Solving Equations - When you see an equation like |x - 3| = 7, split it into two scenarios: x - 3 = 7 and x - 3 = - 7. That way you find both solutions (x = 10 or x = - 4) and nail every possible case. Absolute Value Equations
  4. Graphing the "V" Shape - Plotting y = |x| gives you a distinctive V‑shape, thanks to its piecewise nature. It's smooth on each side yet changes direction at the origin, making it a fun graph to master. Absolute Value Graph
  5. Real‑Life Connections - From calculating distances on a map to measuring error margins in experiments, absolute value shows up everywhere. It helps you ignore "direction" and focus just on how big a difference is. Real‑Life Applications
  6. Even Function Insight - Absolute value is an even function, meaning f(x) = f( - x). Its graph is a mirror image on either side of the y‑axis - perfect symmetry for visual learners. Even Function Property
  7. Sign Function Link - You can express |x| as x times the signum function: |x| = x · sgn(x). This nifty relationship ties absolute value directly to the idea of "positive or negative." Sign Function Relationship
  8. Inequality Strategies - To solve something like |x| < 3, break it into - 3 < x < 3. For |x - 2| ≥ 5, consider x - 2 ≥ 5 or x - 2 ≤ - 5. Casework is your secret weapon! Inequality Techniques
  9. Complex Number Extension - For a complex number a + bi, its absolute value is √(a² + b²), the distance from the origin in the complex plane. It merges algebra and geometry in a super cool way. Complex Absolute Value
  10. Practice Makes Perfect - Drill yourself with a mix of equations, inequalities, and graphing problems to build confidence. The more you play with absolute values, the more natural they'll feel when you hit tougher topics. Practice Problems & Examples
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