Section 2: Solving Absolute Value Equations

Solving Absolute Value Equations of the Type | x | = k.

Absolute value equations are useful in determining distance and error measurements.

Example 1 : Solve for x:  | x | = 3

Solution.

This equation is asking us to find all numbers, x, that are 3 units from zero on the number line.
We must consider numbers both to the right and to the left of zero on the number line.

Notice that both 3 and -3 are three units from zero. 
The solution is:   x = 3  or  x = −3.
 

Example 1 suggests a rule that we can use when solving absolute value equations.

If c is a positive number, then | x | = c is equivalent to  x = c or x = –c.

Example 2 : Solve for x:  | x – 6 | = 4

Solution.

 
Step 1. Break the equation up into two equivalent equations using the rule:  If |x| = c then x = c or x = -c.  

  | x – 6 | = 4  is equivalent to x – 6 = 4     or     x – 6 = – 4
 
Step 2. Solve each equation. 

Step 3. Check the solutions.  

  | 10 – 6 | = | 4 | = 4
 
  | 2 – 6 | = | –4 | = 4
 
The solutions are x = 10 and x = 2.
 

Example 3 : Solve for x:  | 2x – 3 | = 9

Solution.

Step 1. Break the equation up into two equivalent equations using the rule:  If |x| = c then x = c or x = -c.  

 | 2x – 3 | = 9 is equivalent to  2x – 3 = 9   or   2x – 3  = -9

Step 2. Solve each equation. 

2x – 3 = 9   or   2x – 3  = -9

2x – 3 + 3 = 9 + 3   or   2x – 3 + 3 = -9 + 3

2x = 12 or 2x  = -6

2x ÷ 2 = 12 ÷ 2  or  2x ÷ 2 = -6 ÷ 2

x = 6   or  x  = -3

 
Step 3. Check the solutions.  

x = 6: | 2(6) – 3 | = | 12 – 3  | = | 9 | = 9
 
x = -3: | 2(-3) – 3 | = | -6 – 3  | = | -9 | = 9

The solutions are x = 6 and x = -3

Example 4 : Solve for x:  | x + 7 | = –2

Solution.

The absolute value of a number is never negative.  This equation has no solution.

Solving Absolute Value Equations of the Type | x | = | y |.

If the absolute values of two expressions are equal, then either the two expressions are equal, or they are opposites.

If x and y represent algebraic expressions, | x | = | y | is equivalent to  x = y or x = –y.

Example 5 : Solve for x:  | x + 8 | = | 3x – 4 |

Solution.

Step 1. Break the equation up into two equivalent equations.

  | x + 8 | = | 3x – 4 | is equivalent to x + 8 = 3x – 4   or   x + 8 = –(3x – 4)

 
Step 2. Solve each equation. 

x + 8 = 3x – 4  or   x + 8 = –(3x – 4)

x + 8 = 3x – 4    or   x + 8 = –3x + 4

x + 8 – x  = 3x – 4 – x      or    x + 8 + 3x = -3x + 4 + 3x

8 = 2x – 4  or   4x + 8 = 4

8 + 4 = 2x – 4 + 4   or   4x + 8 – 8  = 4 – 8

12 = 2x    or   4x  = – 4

12 ÷ 2 = 2x ÷ 2   or   4x÷ 4 = – 4 ÷ 4

6 = x    or    x  = – 1

 
Step 3. Check the solutions.

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