he term "absolute value" comes up in a first year algebra course. But what exactly does absolute value mean? Absolute value is essentially the distance between a number and 0 and since distance is always a positive number, the absolute value of any number is always positive. For example, take two numbers -4 and 4. The distance from -4 to 0 is 4 and the distance from 4 to 0 is also 4, so the absolute value of -4 and 4 are both 4.

Absolute value is noted as follows:

Absolute value of 4... |4| = 4.
Absolute value of -4... |-4| = 4.

When solving equations with absolute value, we have to set up two equations, a positive case and a negative case. For example, |x| = 5 means that the positive case is x = 5 and the negative case is x = -5. The same idea holds true for more complex problems with absolute value.

Example: Solve |x + 5| = 9.

The first equation (positive case) is x + 5 = 9, therefore x = 4. The second equation (negative case) is x + 5 = -9, therefore x = -14.

Example: Solve |2x + 7| = 4x + 2.

Positive case is 2x + 7 = 4x + 2.

2x = 4x + 2 - 7 (subtract 7 from each side)
2x = 4x - 5
2x - 4x = -5 (subtract 4x from each side to isolate x)
- 2x = -5
x = 5/2 (divide both sides by -2)

Negative case is 2x + 7 = -(4x + 2)

2x + 7 = -4x - 2 (apply the - in front of the parentheses on the right side)
2x = -4x - 2 - 7 (subtract 7 from both sides)
2x = -4x - 9
2x + 4x = -9 (add 4x to both sides)
6x = -9
x = -9/6 (divide both sides by 6)
x = -3/2 (simplify -9/6 to -3/2)

Substitute each value for x into the original equation to see which satisfy the equation. There is no need for this substitution on equations without a variable on both sides of the equation. In this case, x = -3/2 will not satisfy the equation and therefore is not a solution.

Example: Solve 3|4x - 5| = 6x.

Positive case is 3(4x - 5) = 6x
12x - 15 = 6x (apply the distributive property)
12x = 6x + 15 (add 15 to both sides)
12x - 6x = 15 (subtract 6x from both sides)
6x = 15 (combine like terms on left side)
x = 15/6 (divide both sides by 6)
x = 5/2 (simplify 15/6 to 5/2)

Negative case is 3(4x - 5) = -6x
12x - 15 = -6x
12x = -6x + 15
12x + 6x = 15
18x = 15
x = 15/18
x = 5/6

When check the possible solutions to make sure they satisfy the original equation, you will see that both satisfy the equation, therefore they are both solutions.

The absolute value inequality will be in the forms |x| < h, |x| ≤ h, |x| > h or |x| ≥ h. To solve such equations, we still have a positive and a negative case. To set up the equations for the positive case, we drop the absolute value sign and rewrite the equation . For the negative case, we drop the absolute value sign, take the negative of the other side and reverse the inequality symbol.

Example: Solve |2x - 5| < 9.

Positive case 2x - 5 < 9
2x < 14
x < 7

Negative case 2x - 5 > -9
2x > -4
x > -2

The solution is the set (-2, 7).

Example: Solve |4x - 3| ≥ 13.

Positive case 4x - 3 ≥ 13
4x ≥ 16
x ≥ 4

Negative case 4x - 3 ≤ -13
4x ≤ -10
x ≤ -10/4
x ≤ -5/2

The solution is (-∞, -5/2] U [4, ∞).

The most important thing to remember is to set up both positive and negative cases. In the negative case with inequalities, remember to flip the inequality sign and negate whatever is after the sign.

This guide should help clear any confusion about absolute value and solving equations and inequalities dealing with absolute value.