Understanding Absolute Value Equations
Welcome to Professor Baker's Math Class! Today, we're tackling absolute value equations. Remember, the absolute value of a number is its distance from zero. This means that an absolute value equation often has two possible solutions, one positive and one negative.
Key Concept: Distance from Zero
The absolute value, denoted by $|x|$, represents the distance of $x$ from zero on the number line. For example:
- $|2| = 2$ because 2 is 2 units away from zero.
- $|-2| = 2$ because -2 is also 2 units away from zero.
Therefore, when solving an absolute value equation like $|x| = a$, we need to consider both $x = a$ and $x = -a$ as possible solutions.
Solving Absolute Value Equations
Let's look at some examples:
Example 1:
Solve $|x| = 5$
This means $x$ is 5 units away from zero. So, we have two possible solutions:
- $x = 5$
- $x = -5$
Thus, the solutions are $x = 5$ and $x = -5$.
Example 2:
Solve $|2x - 9| = 11$
Here, the expression inside the absolute value, $2x - 9$, must be either 11 or -11.
- Case 1: $2x - 9 = 11$
Add 9 to both sides: $2x = 20$
Divide by 2: $x = 10$ - Case 2: $2x - 9 = -11$
Add 9 to both sides: $2x = -2$
Divide by 2: $x = -1$
So, the solutions are $x = 10$ and $x = -1$.
Example 3:
Solve $|5 - 2x| = 13$
- Case 1: $5 - 2x = 13$
Subtract 5 from both sides: $-2x = 8$
Divide by -2: $x = -4$ - Case 2: $5 - 2x = -13$
Subtract 5 from both sides: $-2x = -18$
Divide by -2: $x = 9$
The solutions are $x=-4$ and $x=9$
Important Note
Absolute value cannot be negative. So, if you encounter an equation like $|x| = -2$, there is no solution.
Practice Problems
Your homework is to practice these skills! Please complete problems Pg. 53 #32-40. Keep practicing, and you'll master absolute value equations in no time!