How to Solve Absolute Value Equations

How to Solve Absolute Value Equations

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Absolute value equations show up in everything from engineering to finance—miss them on your exam, and you’re leaving easy points on the table. Today, you’ll learn the exact steps to solve them every single time."

What You Need To Know First

Before diving in, make sure you understand: 1. What absolute value means – The distance from zero on a number line, always non-negative. 2. Solving linear equations – You should be comfortable solving equations like 2x + 3 = 7. 3. Checking solutions – Plugging answers back into the original equation to verify.

If any of these feel shaky, pause and review them first.

Key Vocabulary

Formulas To Know

1. Definition of Absolute Value

Formula: |A| = - A if A ≥ 0 - -A if A < 0

What it means: - If the expression inside the absolute value is zero or positive, the absolute value is the expression itself. - If the expression inside is negative, the absolute value is its opposite.

Memorise This. – It’s the foundation of solving absolute value equations.

2. Solving Absolute Value Equations

Formula: If |A| = B, then: - A = B OR A = -B

Conditions: - B must be ≥ 0 (absolute value can’t equal a negative number). - If B < 0, there’s no solution.

Memorise This. – This is the core method for solving.

Step-by-Step Method

Follow these steps exactly for every absolute value equation.

Step 1: Isolate the Absolute Value

• Move everything outside the absolute value to the other side of the equation.
• Goal: Get |expression| = number.

Example: Solve |2x + 3| - 5 = 10 → Add 5 to both sides: |2x + 3| = 15

Step 2: Check if the Right Side is Negative

• If the number on the right is negative, stop here—there’s no solution.
• If it’s zero or positive, proceed to Step 3.

Example: |2x + 3| = -4 → No solution (absolute value can’t be negative).

Step 3: Split into Two Cases

• Write two separate equations using the definition of absolute value:
• expression = right side
• expression = -(right side)

Example: |2x + 3| = 15 → 1. 2x + 3 = 15 2. 2x + 3 = -15

Step 4: Solve Both Equations

• Solve each equation separately using standard algebra.

Example: 1. 2x + 3 = 15 → 2x = 12 → x = 6 2. 2x + 3 = -15 → 2x = -18 → x = -9

Step 5: Check for Extraneous Solutions

• Plug both solutions back into the original equation.
• If a solution doesn’t work, cross it out.

Example: Check x = 6: |2(6) + 3| - 5 = |15| - 5 = 10 ✔️ Check x = -9: |2(-9) + 3| - 5 = |-15| - 5 = 10 ✔️ Both work.

Step 6: Write the Final Answer

• List all valid solutions.
• If no solutions work, write "No solution."

Example: x = 6 or x = -9

Worked Examples

Example 1 – Basic (No Tricks)

Problem: Solve |x - 4| = 7

Step 1: Absolute value is already isolated. Step 2: Right side (7) is positive → proceed. Step 3: Split into two cases: 1. x - 4 = 7 2. x - 4 = -7 Step 4: Solve both: 1. x = 11 2. x = -3 Step 5: Check both: - |11 - 4| = 7 ✔️ - |-3 - 4| = 7 ✔️ Step 6: Final answer: x = 11 or x = -3

What we did and why: We split the equation into two cases because absolute value can be either positive or negative. Both solutions worked, so we kept them.

Example 2 – Medium (Added Complication)

Problem: Solve 3|2x + 1| - 4 = 8

Step 1: Isolate the absolute value: 3|2x + 1| = 12 → |2x + 1| = 4 Step 2: Right side (4) is positive → proceed. Step 3: Split into two cases: 1. 2x + 1 = 4 2. 2x + 1 = -4 Step 4: Solve both: 1. 2x = 3 → x = 1.5 2. 2x = -5 → x = -2.5 Step 5: Check both: - 3|2(1.5) + 1| - 4 = 3|4| - 4 = 8 ✔️ - 3|2(-2.5) + 1| - 4 = 3|-4| - 4 = 8 ✔️ Step 6: Final answer: x = 1.5 or x = -2.5

What we did and why: We first isolated the absolute value by dividing both sides by 3. Then, we split and solved as usual. Both solutions checked out.

Example 3 – Exam Style (Disguised Problem)

Problem: Solve |5 - 2x| = x + 1

Step 1: Absolute value is isolated. Step 2: Right side must be ≥ 0 → x + 1 ≥ 0 → x ≥ -1 Step 3: Split into two cases: 1. 5 - 2x = x + 1 2. 5 - 2x = -(x + 1) Step 4: Solve both: 1. 5 - 2x = x + 1 → 4 = 3x → x = 4/3 ≈ 1.33 2. 5 - 2x = -x - 1 → 6 = x → x = 6 Step 5: Check both (and ensure x ≥ -1): - x = 4/3: |5 - 2(4/3)| = 7/3 vs. 4/3 + 1 = 7/3 ✔️ - x = 6: |5 - 2(6)| = 7 vs. 6 + 1 = 7 ✔️ Step 6: Final answer: x = 4/3 or x = 6

What we did and why: We had to check the right side first (x + 1 ≥ 0) because absolute value can’t equal a negative. Both solutions worked, so we kept them.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. Absolute value equations always follow the same steps: 1. Isolate the absolute value—get it by itself. 2. Check if the right side is negative. If it is, no solution. 3. Split into two cases: one positive, one negative. 4. Solve both equations. 5. Check your answers in the original equation—extraneous solutions are sneaky! 6. Write your final answer.

Remember: Absolute value can’t be negative, so if you see |x| = -5, stop immediately—no solution. And if the right side has a variable, like |x + 2| = 3x - 1, check if 3x - 1 ≥ 0 first.

You’ve got this. Now go practice—try one with a fraction, one with a negative, and one where you have to isolate first. See you in the next one!