lesson 7.2 writing two step inequalities answer key

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23-11-2024

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Lesson 7.2: Writing Two-Step Inequalities – A Comprehensive Guide

This article serves as a guide to understanding and solving two-step inequalities, a crucial topic in algebra. We'll break down the process step-by-step, providing examples and addressing common challenges. While I cannot provide a specific "answer key" for a particular textbook's Lesson 7.2, I can equip you with the tools to solve any problem you encounter. Remember to always consult your textbook and teacher for specific exercises and solutions related to your curriculum.

Understanding Inequalities

Before tackling two-step inequalities, let's review the basics. Inequalities compare two expressions, indicating that one is greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) the other. Unlike equations, inequalities have multiple solutions.

What are Two-Step Inequalities?

Two-step inequalities involve two operations performed on a variable. These operations could be addition, subtraction, multiplication, or division. The goal is to isolate the variable to find the range of values that satisfy the inequality.

Solving Two-Step Inequalities: A Step-by-Step Approach

Solving two-step inequalities is similar to solving two-step equations, but with one crucial difference: when multiplying or dividing by a negative number, you must reverse the inequality sign.

Step 1: Simplify

First, simplify both sides of the inequality by combining like terms if possible.

Example: 3x + 5 > 11

Step 2: Isolate the Term with the Variable

Perform the inverse operation to remove any constants added to or subtracted from the term containing the variable.

Example (continued): Subtract 5 from both sides: 3x > 6

Step 3: Isolate the Variable

Perform the inverse operation to isolate the variable completely. Remember to reverse the inequality sign if you multiply or divide by a negative number.

Example (continued): Divide both sides by 3: x > 2

Step 4: Check Your Solution

Select a value within the solution set and substitute it back into the original inequality. If the inequality holds true, your solution is correct.

Example (continued): Let's test x = 3: 3(3) + 5 > 11 => 14 > 11 (True)

Common Mistakes to Avoid

• Forgetting to Reverse the Inequality Sign: Remember to reverse the inequality symbol when multiplying or dividing by a negative number. This is a very common error.

• Incorrect Order of Operations: Follow the order of operations (PEMDAS/BODMAS) carefully.

• Sign Errors: Be meticulous with your positive and negative signs. A simple sign error can lead to an incorrect solution.

Examples of Two-Step Inequalities

Let's work through a few more examples:

Example 1:

-2x + 7 ≤ 1

1. Subtract 7 from both sides: -2x ≤ -6
2. Divide both sides by -2 and reverse the inequality sign: x ≥ 3

Example 2:

5x - 3 > 12

1. Add 3 to both sides: 5x > 15
2. Divide both sides by 5: x > 3

Example 3: Involving Fractions

(1/2)x + 4 < 10

1. Subtract 4 from both sides: (1/2)x < 6
2. Multiply both sides by 2: x < 12

Graphing Inequalities

Once you've solved the inequality, you can represent the solution on a number line. Use an open circle (o) for > or < and a closed circle (•) for ≥ or ≤.

How to Tackle Word Problems

Many real-world scenarios can be modeled using two-step inequalities. Carefully read the problem, identify the unknown variable, and translate the words into mathematical symbols.

Conclusion:

Mastering two-step inequalities is a crucial skill in algebra. By following these steps and practicing regularly, you can confidently solve any two-step inequality problem you encounter. Remember to always double-check your work and understand the reasoning behind each step. If you're still struggling with specific problems from Lesson 7.2 in your textbook, consult your teacher or classmates for assistance. Consistent practice is key to success!