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

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

The examples that we will consider are:

| *x* | = 3

| *x* – 6 | = 4

| 2 *x* – 3 | = 9

| x + 7 | = ** –**2

| *x*+ 8 | = | 3*x* – 4 |

* 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*= −3Example 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

**or**

*x*– 6 = 4

*x*– 6 = – 4**Step 2.** **Solve each equation** .

*x* – 6 + 6 **= 4 + 6**

** x = 10 **

** x – 6 + 6 = – 4 + 6 **

** x = 2 **

**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

**: | 2**

*x**x*– 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 . **

** | 2 x – 3 | = 9 ** is equivalent to

**2**or

*x*– 3 = 9**2**

*x*– 3 = -9**Step 2.** **Solve each equation** .

** 2 x – 3 = 9 ** or

**2**

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

**2**

*x*– 3 + 3 = -9 + 3** 2 x = 12 ** or

**2**

*x*= -6** 2 x ÷ 2 = 12 ÷ 2 ** or

**2**

*x*÷ 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 | = | 3*x*– 4 |** Solution. **

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

** | x + 8 | = | 3 x – 4 | ** is equivalent to

**or**

*x*+ 8 = 3*x*– 4

*x*+ 8 =**(3***–**x*– 4)**Step 2. Solve each equation.**

** x + 8 = 3 x – 4 ** or

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

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

*x*+ 8 + 3*x*= -3*x*+ 4 + 3*x*** 8 = 2 x – 4 ** or

**4**

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

**4**

*x*+ 8 – 8 = 4 – 8** 12 = 2 x ** or

**4**

*x*= – 4** 12 ÷ 2 = 2 x ÷ 2 ** or

**4**

*x*÷ 4 = – 4 ÷ 4** 6 = x ** or

*x*= – 1**Step 3** **.** **Check the solutions.**

*x* = 6: | 6 + 8 | = | 3(6) – 4 |

*|* 14 | = | 18 – 4 |

| 14 | = | 14 |

14 = 14

*x* = ** – ** 1: |

**1 + 8 | = | 3(**

*–***1) – 4 |**

*–*| 7 | = | ** – ** 3 – 4 |

| 7 | = | ** – ** 7 |

7 = ** – ** 7

The solutions are ** x = 6 ** and x

**= – 1**.

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