Solving Absolute Value Equations

Multiple Choice

What's a way to define absolute value?

• The value of a number

• The opposite of a number

• The distance of a number from zero

• The multiplicative inverse of a number

Multiple Choice

What is the reason that absolute value is always written as a positive?

• Absolute value is talking about numbers so it must be positive.

• Absolute value does not always have to be positive.

• Absolute value is like a clock it has only positive numbers.

• Absolute value is talking about distance, distance is always measured by positive numbers.

Absolute value |a|

Absolute value represents a number's distance from zero on a number line. Because it represents a distance, an absolute value is positive.

|-2| = 2 Because -2 is two units away from zero on a number line.

|7| = 7 Because 7 is seven units away from zero on a number line.

Evaluating with Absolute Value

When evaluating mathematic expressions with absolute value, treat the absolute value bars as a grouping symbol, similar to parentheses.

2 -5|3 + 1| Add the numbers inside the bars first. Take the absolute value of that sum. Then multiply by five and subtract that product from 2.

2-5|4|--->2-5(4)--->2-20--->-18

Solving absolute value equations

When solving absolute value equations, it is important to understand the definition of absolute value.

|x| = 5

When solving the above equation, remember that absolute value is a number's distance from zero on the number line. If |x|=5, what number or number is five units away from zero on the number line? That could be 5 or -5. The equation has two answers: x = 5 and x = -5.

What do you do when there is more to the problem than just |x| = 5?

To solve a bigger problem, like

|x - 4| = 2 Start by making sure the absolute value is isolated. Then, consider the possible values the absolute value could equal. In the problem above, an absolute value equals two. This means we have to know what numbers are 2 units from zero on the number line. (That's 2 or -2.) Write two equations that show the possible situations.

x - 4 = -2 .and. x - 4 = 2 Then, solve these smaller equations.

x = 2 .and. x = 6

Remember to plug these back in and make sure they work.

The 4 Steps

When solving an absolute value equation, there's a four step process to consider.

1. Isolate the absolute value
2. Write two equations
3. Solve the two equations
4. Check your answers (sometimes one or both values do not work)