Question

1-Variable Equations & Inequalities Unit Test

Answer 1

I can't provide a test for you directly, but I can certainly help you review concepts related to one-variable equations and inequalities. Here are some key concepts and practice questions you might find useful for studying:

### Concepts to Review:

1. **One-Variable Equations:**
- Standard form: Ax + B = C
- Solving for x: Isolate the variable by performing inverse operations (addition, subtraction, multiplication, division).

2. **One-Variable Inequalities:**
- Symbols: <, >, ≤, ≥
- Solving inequalities involves the same steps as equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.
- Graphing inequalities on a number line.

3. **Compound Inequalities:**
- Solving and graphing statements that combine two inequalities (e.g., a < x < b).

4. **Absolute Value Equations and Inequalities:**
- Equations: |x| = a leads to two possible equations: x = a or x = -a.
- Inequalities: |x| < a and |x| > a; both have specific ways to express the solution set.

5. **Word Problems:**
- Translating real-world problems into equations or inequalities.

### Practice Questions:

#### Equations:
1. Solve for x: \( 2x + 3 = 11 \)
2. Solve for x: \( 5(x - 2) = 3x + 4 \)

#### Inequalities:
3. Solve for x: \( 3x - 5 > 4 \)
4. Solve for x: \( -2(x + 3) ≤ 8 \)

#### Compound Inequalities:
5. Solve for x: \( 1 < 3x + 2 < 10 \)

#### Absolute Value:
6. Solve for x: \( |x - 4| = 7 \)
7. Solve for x: \( |2x + 3| > 5 \)

#### Word Problem:
8. A number decreased by 7 is at least 15. What is the number?

### Sample Solutions:

1. **Solution to 1:**
\[
2x + 3 = 11 \implies 2x = 8 \implies x = 4
\]

2. **Solution to 2:**
\[
5(x - 2) = 3x + 4 \implies 5x - 10 = 3x + 4 \implies 2x = 14 \implies x = 7
\]

3. **Solution to 3:**
\[
3x - 5 > 4 \implies 3x > 9 \implies x > 3
\]

4. **Solution to 4:**
\[
-2(x + 3) ≤ 8 \implies x + 3 ≥ -4 \implies x ≥ -7
\]

5. **Solution to 5:**
\[
1 < 3x + 2 < 10 \implies -1 < 3x < 8 \implies -\frac{1}{3} < x < \frac{8}{3}
\]

6. **Solution to 6:**
\[
|x - 4| = 7 \implies x - 4 = 7 \text{ or } x - 4 = -7 \implies x = 11 \text{ or } x = -3
\]

7. **Solution to 7:**
\[
|2x + 3| > 5 \implies 2x + 3 > 5 \text{ or } 2x + 3 < -5 \implies x > 1 \text{ or } x < -4
\]

8. **Solution to 8:**
Let the number be \( x \):
\[
x - 7 ≥ 15 \implies x ≥ 22
\]

Feel free to work through these problems, and let me know if you have any questions or need further clarification on any topics!