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Solving Equations: (lesson 2 of 4)

Solving Absolute Value Equations

Equations That have Absolute Value Sign on One Side

The absolute value equation |ax + b| = c (c ≥ 0) can be solved by rewriting as two linear equations

ax + b = c or ax + b = -c

and then solving each equation separately.

ABSOLUTE VALUES ALWAYS GIVE 2 EQUATIONS!

Example1:

Solve |x| = 2

Solution:

x = 2 or x = -2

Example 2:

Solve |x + 1| = 2

Solution:

x + 1 = 2 or x + 1 = -2

x = 1 or x = -3

Example 3

Solve |3x - 4| = 5

Solution:

3x - 4 = 5 or 3x - 4 = -5

3x = 9 or 3x = -1

x = 3 or x = -1/3

Example 4

Solve |4x + 7| = -3

Solution:

This equation has no solution, since an absolute value cannot be negative.

Example 5 :

Solve |2x - 6| = 0

Solution:

Since positive and negative 0 mean the same thing, we only need one equation

2x - 6 = 0

2x = 6

x = 3

Exercise 1: Solve absolute value equations

Level 1

 $$\color{blue}{\left| {2x - 3} \right| = 5}$$ $x = 4 , x = 1$ $x = - 4 , x = 1$ $x = 4 , x = - 1$ $x = -4 , x = - 1$

Level 2

 $$\color{blue}{\left| {3x + 7} \right| = 1}$$ $x = - 2 , x = - \frac{8}{3}$ $x = - 2 , x = \frac{8}{3}$ $x = 2 , x = - \frac{8}{3}$ $x = 2 , x = \frac{8}{3}$

Equations That have Absolute Value Signs on Both Sides

If we have absolute value signs on both sides of the equation, we can play the same game with two choices as follows.

Example 6:

Solve |3x + 4| = | 2x - 3|

Solution:

3x + 4 = 2x - 3 or 3x + 4 = -(2x - 3)

3x + 4 = 2x - 3 or 3x + 4 = -2x + 3

3x = 2x - 7 or 3x = -2x - 1

x = -7 or 5x = -1

x = -7 or x = -1/5

Exercise 2: Solve absolute value equations

Level 1

 $$\color{blue}{\left| {3x - 1} \right| = \left| {3 + 4x} \right|}$$ $x = - 4 , x = - \frac{2}{7}$ $x = - 4 , x = \frac{2}{7}$ $x = 4 , x = - \frac{2}{7}$ $x = 4 , x = \frac{2}{7}$

Level 2

 $$\color{blue}{\left| {2 - x} \right| = \left| {4 + x} \right|}$$ $x = 1$ $x = 1 , x = - 1$ $x = -1 , x = - 1$ $x = - 1$