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 – (3x + 1) is 3x + 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:

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) | 5x + 7 | = | 2x – 3 |

2) | 5x – 3 | = | x + 4 |

3) – 5 | 7 – x | + 3 = – 17

4) | t – 3 | + 2t = 6

5) | 3 – 2x | + 4 = 5 – 3 | 3 – 2x |

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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
5x – 3 = x + 4 or 5x – 3 = – (x +4)	– 5 | 7 – x | = – 20 so | 7 – x | = 4
4x = 7 or 6x = –1	7 – x = – 4 or 7 – x = 4
x = 7/4 or 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 t = 3 for both situations		x = 13 /8 or x = 11/8

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