Lesson 7.2 Writing Two Step Inequalities Answer Key

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

This comprehensive guide delves into the intricacies of writing two-step inequalities, a crucial concept in algebra. We'll explore the fundamentals, provide numerous examples with detailed explanations, and offer strategies for mastering this skill. This guide serves as more than just an "answer key"; it's a complete learning resource designed to enhance your understanding and problem-solving abilities.

Understanding Inequalities

Before tackling two-step inequalities, let's solidify our understanding of inequalities in general. Inequalities compare two expressions, indicating that they are not equal. We use the following symbols:

• <: Less than
• >: Greater than
• ≤: Less than or equal to
• ≥: Greater than or equal to

Unlike equations (which use an equals sign), inequalities represent a range of possible values. For example, x > 5 means x can be any number greater than 5, while y ≤ 10 means y can be 10 or any number less than 10.

Deconstructing Two-Step Inequalities

A two-step inequality involves two operations performed on the variable. These operations can include addition, subtraction, multiplication, and division. The goal remains the same as with equations: to isolate the variable on one side of the inequality sign.

Key Principles:

• Inverse Operations: Use inverse operations to undo the operations performed on the variable. Addition and subtraction are inverse operations, as are multiplication and division.
• Maintaining Balance: Whatever operation you perform on one side of the inequality, you must perform on the other side to maintain balance.
• Reversing the Inequality Sign: If you multiply or divide both sides of the inequality by a negative number, you must reverse the direction of the inequality sign. This is crucial and often overlooked.

Solving Two-Step Inequalities: Step-by-Step Process

Let's break down the process with a detailed example:

Example 1: Solve 3x + 5 > 11

1. Isolate the term with the variable: Subtract 5 from both sides: 3x + 5 - 5 > 11 - 5 3x > 6

2. Isolate the variable: Divide both sides by 3: 3x / 3 > 6 / 3 x > 2

Therefore, the solution to the inequality is x > 2. This means x can be any number greater than 2.

Example 2: Solve -2y - 7 ≤ 3

1. Isolate the term with the variable: Add 7 to both sides: -2y - 7 + 7 ≤ 3 + 7 -2y ≤ 10

2. Isolate the variable: Divide both sides by -2. Remember to reverse the inequality sign! -2y / -2 ≥ 10 / -2 y ≥ -5

The solution is y ≥ -5, meaning y can be -5 or any number greater than -5.

Handling Fractions and Decimals

Two-step inequalities can also involve fractions and decimals. The process remains the same, but you may need to perform additional steps to simplify the equation.

Example 3: Solve (1/2)z + 3 < 7

1. Isolate the term with the variable: Subtract 3 from both sides: (1/2)z + 3 - 3 < 7 - 3 (1/2)z < 4

2. Isolate the variable: Multiply both sides by 2: 2 * (1/2)z < 4 * 2 z < 8

Example 4: Solve 0.5a - 2.5 ≥ 1.5

1. Isolate the term with the variable: Add 2.5 to both sides: 0.5a - 2.5 + 2.5 ≥ 1.5 + 2.5 0.5a ≥ 4

2. Isolate the variable: Divide both sides by 0.5: 0.5a / 0.5 ≥ 4 / 0.5 a ≥ 8

Word Problems and Real-World Applications

Two-step inequalities are frequently used to model real-world scenarios. Let's look at an example:

Example 5: Sarah is saving money for a new bicycle that costs $250. She already has $75 saved and earns $15 per week mowing lawns. How many weeks (w) must she work to have enough money for the bicycle?

This can be represented by the inequality: 75 + 15w ≥ 250

1. Solve for w: 15w ≥ 250 - 75 15w ≥ 175 w ≥ 175 / 15 w ≥ 11.67

Since Sarah can't work a fraction of a week, she needs to work at least 12 weeks to have enough money for the bicycle.

Common Mistakes to Avoid

Several common mistakes can hinder your progress when solving two-step inequalities:

• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number. This is a critical error that leads to incorrect solutions.
• Incorrectly applying inverse operations. Ensure you're using the correct inverse operation (addition/subtraction, multiplication/division).
• Losing track of the inequality sign. Keep the inequality sign consistent throughout your calculations.
• Making arithmetic errors. Carefully check your calculations to avoid errors in addition, subtraction, multiplication, and division.

Practice Problems

To solidify your understanding, try solving these problems:

1. 4x - 9 < 7
2. -3y + 6 ≥ 12
3. (2/3)z - 5 ≤ 1
4. 0.75a + 2 > 5.5
5. -1/4 b + 3 < 1

Remember to show your work step-by-step and check your answers carefully.

Advanced Concepts: Compound Inequalities

Once you're comfortable with two-step inequalities, you can move on to compound inequalities, which involve more than one inequality connected by "and" or "or". These require slightly different solution techniques.

Graphing Inequalities

Visualizing the solution to an inequality by graphing it on a number line is a beneficial practice. Remember to use open circles for "<" and ">", and closed circles for "≤" and "≥".

Conclusion

Mastering two-step inequalities is a cornerstone of algebraic proficiency. By understanding the fundamental principles, practicing regularly, and avoiding common pitfalls, you can confidently tackle these problems and apply your skills to solve real-world situations. This guide provides a solid foundation, allowing you to build your algebraic abilities further. Remember to consistently review the concepts and practice solving a wide range of problems to fully grasp this vital mathematical skill. Continued practice is key to achieving mastery and confidence in solving two-step inequalities.