Solving Absolute Value Equations |

**Absolute Value Equations**

Let's first discuss what **absolute value** means. Usually we learn that the absolute value of any number is the positive value of the number but that's not quite true. The words say it all. **The absolute value of a number is the value without regard to sign**. The **signs** we attribute to numbers** indicate a quality** rather than quantity. When I say that the temperature is – 3 degrees, I'm indicating that the temperature is 3 degrees **below zero**.

**The minus sign indicates direction** but has **nothing to do with the** ** value** of the number.

So, when we talk about the absolute value of a number we are talking about **how many** items there are **rather than what direction** or quality that number has. Therefore, to find the absolute value of a number or expression, simply **disregard the sign** before the number or expression.

For instance, the absolute value of – 4 is 4. The absolute value of – (3*x* + 1) is 3*x* + 1.

If we're told that the absolute value of an expression is equal to 5, we know that the expression equals either + 5 or – 5 since both of these numbers have an absolute value of 5.

Absolute value is denoted by vertical lines on either side of the expression as in the

statement | *x* – 3 | = 9, spoken "the absolute value of *x* minus 3 equals 9.

Let's solve this question as an example.

Since we know that the absolute value of *x* – 3 = 9,

we know that *x* – 3 must = either + 9 or – 9, so that's our first step,

then we solve the 2 linear equations for 2 solutions like this:

| *x* – 3 | = 9 turns into *x* – 3 = 9 or *x* – 3 = – 9

if *x* – 3 = 9 then *x* = 12

if *x* – 3 = – 9, then *x* = – 6.

Substituting *x* = 12 and *x* = – 6 satisfies the equation.

When there are two absolute value expression in the equation, we proceed the same way.

Here's an example:

Solve

We rewrite it as two equations:

Now we solve the two equations:

If | lump | = a, then lump = a or lump = – a |

**Note:** The absolute value of an expression cannot be negative,

so | *x* + 7 | = – 4 is not an equation!

Be alert. Watch for these things and don't waste time trying to figure them out.

Just write **no solution** and move on!

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**Practice**

Solve for the unknown in these Absolute Value Equations

1) | 5*x* + 7 | = | 2*x* – 3 |

2) | 5*x* – 3 | = | *x* + 4 |

3) – 5 | 7 – *x* | + 3 = – 17

4) | *t* – 3 | + 2*t* = 6

5) | 3 – 2*x* | + 4 = 5 – 3 | 3 – 2*x* |

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**Solutions**

1) | 5x + 7 | = | 2x – 3 | |
1) | 5x + 7 | = | 2x – 3 | |

5x + 7 = 2x – 3 |
5x + 7 = –2x + 3 |

3x = –10 becomes x = –10/3 |
7x = – 4 becomes– x = 4/7 |

2) | 5x –3 | = | x + 4 | |
3) – 5 | 7 – x | + 3 = – 17 |

5 |
– 5 | 7 – x | = – 20 so | 7 – x | = 4 |

4x = 7 or 6x = –1 |
7 – x = – 4 or 7 – x = 4 |

or x = 7/4–x = 1/6 |
x = 11 or x = 3 |

4) | t – 3 | + 2t = 6 |
5) | 3 – 2x | + 4 = 5 – 3 | 3 – 2x | | ||

| t – 3 | = 6 – 2t |
4 | 3 – 2x | = 1, so | 3 – 2x | = 1/4 | ||

t – 3 = – ( 6 – 2t ) |
or t – 3 = 6 – 2t |
3 – 2x = – 1/4 |
or 3 – 2x = 1/4 |

so for both situationst = 3 |
or x = 13 /8x = 11/8 |

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