Solve Absolute Value Equations
As we prepare to solve absolute value equations, we review our definition of absolute value.
The absolute value of a number is its distance from zero on the number line.
The absolute value of a number n is written as and for all numbers.
Absolute values are always greater than or equal to zero.
We learned that both a number and its opposite are the same distance from zero on the number line. Since they have the same distance from zero, they have the same absolute value. For example:
is 5 units away from 0, so
is 5 units away from 0, so
(Figure) illustrates this idea.
For the equation we are looking for all numbers that make this a true statement. We are looking for the numbers whose distance from zero is 5. We just saw that both 5 and are five units from zero on the number line. They are the solutions to the equation.
The solution can be simplified to a single statement by writing This is read, “x is equal to positive or negative 5”.
We can generalize this to the following property for absolute value equations.
For any algebraic expression, u, and any positive real number, a,
Remember that an absolute value cannot be a negative number.
Solve: ⓐ ⓑ ⓒ
ⓐ
ⓑ
Since an absolute value is always positive, there are no solutions to this equation.
ⓒ
Both equations tell us that and so there is only one solution.
Solve: ⓐ ⓑ ⓒ
ⓐⓑ no solution ⓒ 0
Solve: ⓐ ⓑ ⓒ
ⓐⓑ no solution ⓒ 0
To solve an absolute value equation, we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.
Solve
Solve:
Solve:
The steps for solving an absolute value equation are summarized here.
- Isolate the absolute value expression.
- Write the equivalent equations.
- Solve each equation.
- Check each solution.
Solve
| Isolate the absolute value expression. | ||
| Write the equivalent equations. | or | |
| Solve each equation. | or | |
| Check:
|
Solve:
Solve:
Remember, an absolute value is always positive!
Solve:
Solve:
No solution
Solve:
No solution
Some of our absolute value equations could be of the form where u and v are algebraic expressions. For example,
How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, u, and a positive real number, a, if then or
This tell us that
This leads us to the following property for equations with two absolute values.
For any algebraic expressions, u and v,
When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.
Solve:
Solve:
Solve:
Solve Absolute Value Inequalities with “Less Than”
Let’s look now at what happens when we have an absolute value inequality. Everything we’ve learned about solving inequalities still holds, but we must consider how the absolute