# Solve The Absolute Value Equations Worksheet

## 6 problems

Solving the absolute value equations means finding the values of a variable that make an equation with absolute values true. Absolute value equations look like $$|x| = a$$ or $$|x - b| = c$$, where you consider both positive and negative solutions. For example, $$|x| = 3$$ means $$x$$ can be 3 or -3. Solving these equations helps you understand both parts of the solution.

Algebra 2
Equations

## How Will This Worksheet on "Solve the Absolute Value Equations" Benefit Your Student's Learning?

• Clarify the concept of absolute value.
• Enhance problem-solving abilities.
• Promote logical thinking with positive and negative solutions.
• Provide practice in solving absolute value equations.
• Develop critical thinking skills.
• Build a foundation for advanced algebra topics.
• Increase confidence in solving equations.
• Engage students with diverse math problems.

## How to Solve the Absolute Value Equations?

1. Understand that the absolute value of a number is its distance from zero on the number line, always positive. For example, ∣x∣=a means x can be a or −a.

2. For an equation ∣x∣=a:

• If a is positive, set up two equations: x=a and x=−a.
• If a is zero, then ∣x∣=0 means x=0.
• If a is negative, there is no solution because absolute values cannot be negative.

3. Solve the equations you set up in step 2.

## Solved Example

Q. Solve for $z$.$\newline$$-4 = |z| - 9$$\newline$Write your answers as integers or as proper or improper fractions in simplest form.$\newline$$z =$ _____ or $z =$ _____
Solution:
1. Understand and Isolate: Understand the equation and isolate the absolute value.$\newline$We have the equation $–4 = |z| − 9$.$\newline$ To isolate the absolute value, we need to add $9$ to both sides of the equation.$\newline$$-4 + 9 = |z| − 9 + 9$$\newline$$5 = |z|$
2. Solve for $z$: $\newline$Since $|z| = 5$, $z$ can be either $5$ or $-5$ because the absolute value of both $5$ and $-5$ is $5$.$\newline$So, $z = 5$ or $z=-5$.

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