Absolute Value Practice Problems - Equations & Inequalities

Master absolute value equations and inequalities with step-by-step practice problems. Learn to solve |x+7|=12 type equations and interpret distance from zero.

📚What You'll Master in This Practice Session
  • Calculate absolute values of positive and negative numbers using distance from zero
  • Solve absolute value equations with variables by splitting into two cases
  • Apply the two-case method to equations like |x+7|=12 and find all solutions
  • Interpret absolute value inequalities as distance ranges on the number line
  • Solve inequalities like |x|<5 and express solutions in interval notation
  • Graph absolute value inequality solutions and understand their geometric meaning

Understanding Absolute value

Complete explanation with examples

Absolute Value

Absolute value is denoted by || and represents the distance from zero.
The absolute value of a positive number - will always be the number itself.
For example: │2│=2│2│= 2
The absolute value of a negative number - will always be the same number but positive.
For example: │−3│=3│-3│=3
Note that the absolute value of a number will always be positive since distance is always positive.

Absolute Value in an Equation with a Variable

If we have an unknown or an expression with an unknown inside absolute value, we ask ourselves which expression will give us the desired equation value, split into cases and find the unknown.
For example in the equation: │x+7│=12│x+7│=12
We ask ourselves, which expression in absolute value will equal 12.
The answer will be 12 or -12. (both 12 in absolute value equals 12 and -12 in absolute value equals 12).
Therefore, we'll take the entire expression and split it into two cases:
First case:
x+7=12x+7=12
Let's solve:
x=5x=5

Second case:
X+7=−12X+7=-12
Let's solve:
x=−19x=-19

Therefore, the solution to the exercise is: x=5,−19x=5,-19

Detailed explanation

Practice Absolute value

Test your knowledge with 14 quizzes

\( \left|-2\right|= \)

Examples with solutions for Absolute value

Step-by-step solutions included
Exercise #1

Determine the absolute value of the following number:

∣18∣= \left|18\right|=

Step-by-Step Solution

The "absolute value" can be viewed as the distance of a number from 0.
Therefore, the absolute value will not change the sign from negative to positive, it will always be positive.

Answer:

18 18

Video Solution
Exercise #2

∣−712∣= \left|-7\frac{1}{2}\right|=

Step-by-Step Solution

The absolute value of a number is always its positive value. It represents the distance of the number from zero on the number line, regardless of direction. The absolute value of any negative number is its opposite positive number.

Step 1: Identify the number to find the absolute value of: −712 -7\frac{1}{2}

Step 2: Change the negative sign to positive: 712 7\frac{1}{2}

Hence, the absolute value of −712 -7\frac{1}{2} is 712 7\frac{1}{2} .

Answer:

712 7\frac{1}{2}

Exercise #3

Solve for the absolute value of the following integer:

∣34∣= \left|34\right|=

Step-by-Step Solution

The absolute value of a number is always non-negative because it represents the distance from zero. Therefore, the absolute value of 34 34 is 34 34 .

Answer:

34 34

Exercise #4

∣−7∣= \left|-7\right|=

Step-by-Step Solution

The absolute value of a number is its distance from zero on the number line, regardless of the direction. To find the absolute value of −7 -7 , we need to look at the distance of −7 -7 from zero, which is 7 7 . Therefore, ∣−7∣=7 \left|-7\right| = 7 .

Answer:

7 7

Exercise #5

∣5∣= \left|5\right|=

Step-by-Step Solution

The absolute value of a number is its distance from zero on the number line, without considering its direction. To find the absolute value of 5 5 , consider the distance of 5 5 from zero, which is just 5 5 . Therefore, ∣5∣=5 \left|5\right| = 5 .

Answer:

5 5

Frequently Asked Questions

Everything you need to know about Absolute value

What does absolute value mean in math?

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Absolute value represents the distance of a number from zero on the number line, always expressed as a positive value. It's denoted by vertical bars |x| and tells us how far a number is from zero, regardless of direction.

How do you solve absolute value equations with variables?

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To solve equations like |x+7|=12, split into two cases: (1) x+7=12 and (2) x+7=-12. This is because both 12 and -12 have an absolute value of 12. Solve each case separately to find all possible solutions.

Why does |x|=5 have two solutions?

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The equation |x|=5 asks which numbers are exactly 5 units away from zero. Both x=5 and x=-5 are 5 units from zero, so the complete solution is x=5,-5.

What's the difference between |x|<5 and |x|>5?

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|x|<5 means x is within 5 units of zero, so -55 means x is more than 5 units from zero, so x<-5 or x>5. The first creates one interval, the second creates two separate intervals.

Can absolute value ever be negative?

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No, absolute value is always non-negative (positive or zero). If you encounter an equation like |x|=-3, there is no solution because absolute value represents distance, which cannot be negative.

How do you graph absolute value inequalities?

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For |x|a, shade two rays: x<-a and x>a. Use open circles for < or > and closed circles for ≤ or ≥.

What are common mistakes when solving absolute value equations?

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Common errors include: (1) forgetting the negative case when splitting equations, (2) thinking |x|=-5 has solutions, and (3) incorrectly combining inequality solutions. Always check both cases and verify answers in the original equation.

How is absolute value used in real life?

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Absolute value appears in: measuring temperature differences from a target, calculating distances regardless of direction, determining error margins in measurements, and expressing tolerances in manufacturing. It's essential whenever magnitude matters more than direction.

More Absolute value Questions

Click on any question to see the complete solution with step-by-step explanations

Absolute value

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Continue Your Math Journey

Master Absolute value first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

Absolute ValueNumerical ValueAbsolute Value InequalitiesInequalities with Absolute Value

Practice by Question Type

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