Absolute Value Equations notes worksheet with guided steps and practice problems.

Here are the solutions to the problems on your worksheet. I have worked through each one step-by-step so you can see how to get the answer.

Part 1: Definitions

* absolute value: the distance of a number from zero; the absolute value of a number is always positive (or non-negative).
* example: $|5| = \mathbf{5}$
* example: $|-7| = \mathbf{7}$

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Part 2: Solving Absolute Value Equations

The Example Problem: $|x + 3| + 4 = 9$

* Step 1: Isolate the absolute value expression.
Subtract 4 from both sides:
$$|x + 3| = 5$$
* Step 2: Write the equation twice (positive and negative).
Case A: $x + 3 = 5$
Case B: $x + 3 = -5$
* Step 3: Solve both equations.
Case A: $x = 2$
Case B: $x = -8$

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Part 3: Examples Solutions

1. $|2x - 1| = 11$
* Case 1 (Positive):
$$2x - 1 = 11$$
Add 1 to both sides: $2x = 12$
Divide by 2: $\mathbf{x = 6}$
* Case 2 (Negative):
$$2x - 1 = -11$$
Add 1 to both sides: $2x = -10$
Divide by 2: $\mathbf{x = -5}$

2. $|4x| = 16$
* Case 1 (Positive):
$$4x = 16$$
Divide by 4: $\mathbf{x = 4}$
* Case 2 (Negative):
$$4x = -16$$
Divide by 4: $\mathbf{x = -4}$

3. $|x + 4| - 10 = 3$
* First, isolate the absolute value by adding 10 to both sides:
$$|x + 4| = 13$$
* Case 1 (Positive):
$$x + 4 = 13$$
Subtract 4: $\mathbf{x = 9}$
* Case 2 (Negative):
$$x + 4 = -13$$
Subtract 4: $\mathbf{x = -17}$

4. $|x| - 2 = 16$
* First, isolate the absolute value by adding 2 to both sides:
$$|x| = 18$$
* Since $|x|$ just means the distance of $x$ from zero, the answers are simply:
$\mathbf{x = 18}$ and $\mathbf{x = -18}$

5. $|x| = -5$
* Absolute value represents distance, and distance cannot be negative. There is no number that has a negative distance from zero.
* Answer: No Solution

6. $|\frac{x}{5}| + 2 = 6$
* First, isolate the absolute value by subtracting 2 from both sides:
$$|\frac{x}{5}| = 4$$
* Case 1 (Positive):
$$\frac{x}{5} = 4$$
Multiply by 5: $\mathbf{x = 20}$
* Case 2 (Negative):
$$\frac{x}{5} = -4$$
Multiply by 5: $\mathbf{x = -20}$

Final Answer:
Definitions: distance, zero, positive. Examples: 5, 7.
Main Example: x = 2, x = -8
1. x = 6, x = -5
2. x = 4, x = -4
3. x = 9, x = -17
4. x = 18, x = -18
5. No Solution
6. x = 20, x = -20