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Quizzes > High School Quizzes > Mathematics

Absolute Value Worksheets Practice Quiz

Solve Equations, Inequalities, and Integer Challenges

Difficulty: Moderate
Grade: Grade 7
Study OutcomesCheat Sheet
Colorful paper art promoting an engaging high school algebra quiz on absolute values.

What is the absolute value of -7?
0
14
-7
7
Absolute value measures the distance from zero on the number line. Since distance is always nonnegative, the absolute value of -7 is 7.
What is the absolute value of 5?
-5
10
5
0
The absolute value of a positive number remains the same. Therefore, |5| is equal to 5 directly.
Which of the following is equivalent to |-3|?
3
-3
6
0
Absolute value extracts the nonnegative magnitude of a number. Hence, |-3| equals 3 because it represents the distance from zero.
How is absolute value defined?
The distance a number is from zero on the number line
The negative of a number
The sum of a number and its opposite
A number's square
Absolute value is defined as the distance of a number from zero regardless of its sign. This definition correctly captures the essence of absolute value.
What is the value of |0|?
0
-1
1
Undefined
Zero is exactly at the center of the number line, meaning it has no distance from itself. Therefore, the absolute value of 0 is 0.
Solve the equation |x - 2| = 5. What are the possible values of x?
x = -5 or x = 3
x = -3 or x = 7
x = 2 or x = 5
x = -7 or x = 3
To solve |x - 2| = 5, set x - 2 = 5 and x - 2 = -5. This leads to the solutions x = 7 and x = -3, which are the only values satisfying the equation.
Solve the inequality |x - 1| < 3. Which of the following represents the solution set?
x < -2 or x > 4
x > -2 and x < 4
x > -4 and x < 2
x > 2 and x < 4
The inequality |x - 1| < 3 means that the distance between x and 1 is less than 3. By rewriting it as -3 < x - 1 < 3, we obtain -2 < x < 4 as the solution.
Solve the equation |3 - x| = 1. What values of x satisfy the equation?
x = 0 or x = 1
x = -4 or x = -2
x = 2 or x = 4
x = 1 or x = 3
Setting 3 - x equal to 1 gives x = 2, and setting it equal to -1 gives x = 4. Both x = 2 and x = 4 satisfy the original equation.
Solve the equation |2x| = 10. What are the solutions for x?
x = 2 or x = -2
x = 5 or x = -5
x = 10 or x = -10
x = 0 or x = 10
The equation |2x| = 10 splits into 2x = 10 and 2x = -10. Solving these gives x = 5 and x = -5 respectively, which are the only valid solutions.
Evaluate the expression |-2 + 3|.
5
0
-1
1
First, calculate -2 + 3 to get 1. The absolute value of 1 is 1, making it the correct evaluation of the expression.
Find the solutions to the equation |x + 4| = 3.
x = 3 or x = -3
x = -3 or x = 3
x = 1 or x = 7
x = -1 or x = -7
Setting x + 4 equal to 3 gives x = -1, and setting it equal to -3 gives x = -7. These two values are the complete set of solutions for the equation.
Solve the equation |5 - x| = 0. What is the value of x?
x = 5
x = -5
x = 0
x = 10
An absolute value equals zero only when the expression inside is zero. Therefore, solving 5 - x = 0 yields x = 5, which is the only solution.
Which of the following is the correct piecewise definition of |x - 2|?
x - 2 if x ≥ 2; 2 - x if x < 2
2 + x if x ≥ 2; -2 - x if x < 2
2 - x if x ≥ 2; x - 2 if x < 2
x + 2 if x ≥ 2; -x - 2 if x < 2
For x greater than or equal to 2, the expression (x - 2) is nonnegative, so that is |x - 2|. When x is less than 2, the absolute value is computed as 2 - x. This piecewise definition correctly represents the function.
Solve the equation |2x - 6| = 8. What are the solutions for x?
x = 6 or x = 0
x = 8 or x = -2
x = 7 or x = -1
x = 1 or x = 5
The equation splits into two cases: 2x - 6 = 8 and 2x - 6 = -8. Solving these gives x = 7 and x = -1 respectively, which are the only correct solutions.
Determine the solution to the equation |x| = -3.
x = 3
No solution
x = -3
x = 0
The absolute value of any real number cannot be negative. Since no number satisfies |x| = -3, the equation has no solution.
Solve the equation |x - 3| + |x + 1| = 8. What are the values of x?
x = 3 or x = -1
x = 5 or x = -3
x = 8 or x = 0
x = 4 or x = -4
By splitting the equation into different cases based on the sign of the expressions inside the absolute values, one finds the equation yields x = 5 and x = -3. Substituting these values back confirms that they satisfy the original equation.
Solve the inequality |2x + 1| ≥ 7. Which of the following describes the solution set?
x ≥ 3 or x ≤ -4
x > 3 and x < -4
x ≤ 3
x ≥ -4
The inequality splits into two cases: 2x + 1 ≥ 7, which simplifies to x ≥ 3, and 2x + 1 ≤ -7, which simplifies to x ≤ -4. The union of these cases gives the complete solution set.
Solve the inequality |3x - 4| > 5. What are the solution intervals for x?
x ≥ 3 and x ≤ -1/3
x > 3 or x < -1/3
x > -1/3 and x < 3
x < 3 and x > -1/3
The inequality |3x - 4| > 5 breaks into 3x - 4 > 5 and 3x - 4 < -5. Solving these gives x > 3 and x < -1/3 respectively. The final answer is the union of these two intervals.
Solve the equation |x + 2| = |2x - 1|. What are the possible values of x?
x = 3 or x = -1/3
x = 0 or x = 3
x = -2 or x = 2
x = 1 or x = -1
Considering the two cases, first set x + 2 equal to 2x - 1 and then x + 2 equal to -(2x - 1). These yield x = 3 and x = -1/3 respectively, which are the valid solutions.
Solve the equation |2x + 3| = 4 - x, considering any necessary restrictions. What are the solutions?
x = 7 or x = -1/3
x = 3 or x = -3
x = 1/3 or x = -7
x = 1 or x = -4
First, note that 4 - x must be nonnegative, establishing the domain restriction x ≤ 4. By considering the cases where 2x + 3 is nonnegative and negative separately, the equation produces the solutions x = 1/3 and x = -7, which satisfy all the conditions.
0
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Study Outcomes

  1. Apply the definition of absolute value to solve equations and inequalities.
  2. Analyze expressions by simplifying and evaluating absolute values.
  3. Demonstrate the use of absolute value in interpreting distances on a number line.
  4. Model real-world scenarios using absolute value concepts.

Absolute Value Worksheets Cheat Sheet

  1. Plot absolute value like a distance marker - Absolute value measures how far a number is from zero on the number line, always producing a non-negative result. Picture |−5| = 5 as if you're 5 steps away from zero, whether you start at −5 or 5. It's a simple but crucial concept that you'll use in tons of problems! Wikipedia: Absolute value
  2. Master the four key properties - Absolute values follow some neat rules: they're never negative, only zero at zero (positive-definiteness), multiply just like regular numbers, and satisfy the triangle inequality (|a + b| ≤ |a| + |b|). Get comfortable with these properties and you'll breeze through proofs and problem checks. Wikipedia: Absolute value properties
  3. Solve equations with ± splits - To tackle an equation like |x − 3| = 7, break it into x − 3 = 7 and x − 3 = −7, giving x = 10 or x = −4. This two-worlds approach ensures you find all valid solutions. Keep practicing so you can set up these splits in your sleep! OpenStax: Solve absolute value equations
  4. Convert "less than" inequalities into compounds - For |x − 3| < 7, rewrite it as −7 < x − 3 < 7, then solve to get −4 < x < 10. This compound strategy transforms an inequality into a familiar bound-check exercise. Visual learners can sketch it on a number line for extra clarity. OpenStax: Solve "less than" inequalities
  5. Split "greater than" inequalities into two cases - When you see |x − 3| > 7, split into x − 3 > 7 or x − 3 < −7, yielding x > 10 or x < −4. This two-branch approach highlights the fact that solutions live outside a central interval. It's like guarding two flanks of a fortress! OpenStax: Solve "greater than" inequalities
  6. Apply to real-world tolerances - In manufacturing, tolerances ensure parts fit perfectly: |x − 10| ≤ 0.1 means x can be between 9.9 cm and 10.1 cm. Absolute value elegantly captures "acceptable deviations" around a target. Imagine a precision engineer high-fiving math for this neat trick! GeeksforGeeks: Real-life applications
  7. Sketch solutions on a number line - Drawing number lines for |x − 3| < 7 or > 7 helps you see which intervals light up. Visualizing these regions cements your understanding of distance-based conditions. Plus, it's a great way to show off your doodling skills in math class! GeeksforGeeks: Absolute value inequalities
  8. Remember the two-solution pattern - Every absolute value equation |expression| = k yields two possible cases (positive and negative), so expect two answers whenever k > 0. This dual-answer feature is your golden ticket to checking all solution branches. Embrace the math detective in you! OpenStax: Two-solution equations
  9. Flip the sign when multiplying negatives - If you multiply or divide an absolute value inequality by a negative, don't forget to flip the inequality sign to keep the solution accurate. This rule saves many students from sneaky mistakes during exams. Think of it as a plot twist in your inequality story! EdBoost: Inequality sign rules
  10. Build confidence through practice - The more absolute value exercises you conquer, the more intuitive they become. Try mixed problems - from simple equations to real-world scenarios - to flex your math muscles. Soon you'll tackle quizzes and exams with a grin! OpenStax: Practice problems
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