Algebra 1: Algebra 1 - Solving Absolute Value

Understanding Absolute Value ➕➖

Absolute value represents the distance of a number from zero on the number line. It is always non-negative. The absolute value of a number x is denoted as |x|.

Solving Absolute Value Equations 📝

To solve an absolute value equation of the form |ax + b| = c, where a, b, and c are constants and c ≥ 0, you need to consider two cases:

• Case 1: ax + b = c
• Case 2: ax + b = -c

Solve each case separately to find all possible solutions for x.

Step-by-Step Guide 🪜

1. Isolate the Absolute Value: Make sure the absolute value expression is isolated on one side of the equation.
2. Set Up Two Equations: Create two separate equations, one where the expression inside the absolute value is equal to the positive value on the other side, and another where it's equal to the negative value.
3. Solve Each Equation: Solve each equation for the variable.
4. Check Your Solutions: Substitute each solution back into the original absolute value equation to ensure it is valid. Absolute value equations can sometimes have extraneous solutions.

Example 1: |x - 3| = 5 💡

Solve the equation |x - 3| = 5.

• Case 1: x - 3 = 5
• Case 2: x - 3 = -5

Solving Case 1:

x - 3 = 5
x = 5 + 3
x = 8

Solving Case 2:

x - 3 = -5
x = -5 + 3
x = -2

Checking the solutions:

• For x = 8: |8 - 3| = |5| = 5 (Valid)
• For x = -2: |-2 - 3| = |-5| = 5 (Valid)

Therefore, the solutions are x = 8 and x = -2.

Example 2: |2x + 1| = 7 💡

Solve the equation |2x + 1| = 7.

• Case 1: 2x + 1 = 7
• Case 2: 2x + 1 = -7

Solving Case 1:

2x + 1 = 7
2x = 7 - 1
2x = 6
x = 3

Solving Case 2:

2x + 1 = -7
2x = -7 - 1
2x = -8
x = -4

Checking the solutions:

• For x = 3: |2(3) + 1| = |7| = 7 (Valid)
• For x = -4: |2(-4) + 1| = |-7| = 7 (Valid)

Therefore, the solutions are x = 3 and x = -4.

Extraneous Solutions ⚠️

Sometimes, when solving absolute value equations, you may encounter extraneous solutions. These are solutions that you obtain through the algebraic process but do not satisfy the original equation. Always check your solutions!

Practice Makes Perfect 💪

Keep practicing with different absolute value equations to improve your skills. Remember to isolate the absolute value, set up two cases, solve each case, and check your solutions.