Algebra 1 Practice - Solve and Graph a Two-Step Inequality (Example 1)

### Algebra 1 Practice: Solve and Graph a Two-Step Inequality

Solving and graphing a two-step inequality in Algebra 1 involves applying multiple operations to isolate the variable and then representing the solution set visually on a number line. Let's break down the process:

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#### **Steps to Solve a Two-Step Inequality**

1. **Identify the Inequality:**
- Recognize the inequality form, typically involving two operations, such as \( ax + b \) is less than \( c \) or \( dx - e \) is greater than or equal to \( f \).

2. **Isolate the Variable:**
- Perform the inverse operations to isolate the variable.
- **Undo Addition/Subtraction:** If the variable is added to or subtracted by a number, perform the opposite operation on both sides.
- **Undo Multiplication/Division:** If the variable is multiplied or divided by a number, perform the opposite operation on both sides. Remember, if you multiply or divide by a negative number, reverse the inequality sign.

3. **Solve the Inequality:**
- After isolating the variable, simplify to find the inequality solution.

4. **Example:**
- Solve \( 2x + 3 \) is greater than or equal to \( 7 \):
1. Subtract 3 from both sides: \( 2x \) is greater than or equal to \( 7 - 3 \)
2. Simplify: \( 2x \) is greater than or equal to \( 4 \)
3. Divide both sides by 2: \( x \) is greater than or equal to \( 4 \div 2 \)
4. Simplify: \( x \) is greater than or equal to \( 2 \)

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#### **Steps to Graph the Inequality on a Number Line**

1. **Draw a Number Line:**
- Create a number line that includes the critical values found in the inequality.

2. **Mark the Critical Values:**
- Identify the critical values on the number line.

3. **Determine the Type of Circle:**
- Use an **open circle** (○) or a **closed circle** (●) based on whether the boundary value is included in the solution or not.

4. **Shade the Region:**
- Shade the region on the number line that satisfies the inequality.

5. **Example:**
- For \( x \) is greater than or equal to 2:
- Draw a number line.
- Mark 2 with a closed circle (●) because \( x \) is greater than or equal to 2.
- Shade to the right of 2.

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#### **Complete Example: Solving and Graphing**

- **Inequality:** \( 3x - 4 \) is less than 5
1. **Solve:**
- Add 4 to both sides: \( 3x \) is less than \( 5 + 4 \)
- Simplify: \( 3x \) is less than 9
- Divide both sides by 3: \( x \) is less than \( \frac{9}{3} \)
- Simplify: \( x \) is less than 3
2. **Graph:**
- Draw a number line.
- Mark 3 with an open circle (○) because \( x \) is less than 3.
- Shade to the left of 3.

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By following these steps, you can effectively solve and graph two-step inequalities, visually representing the solution set on a number line. This process helps in understanding and interpreting the solutions for various inequality problems in Algebra 1. Regular practice with different types of inequalities will help reinforce these skills.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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