Lesson 4.4: Linear Inequalities

Introduction

In this lesson, we will explore linear inequalities, an essential concept in algebra that helps us understand and describe relationships between quantities. By the end of this lesson, students, you will be able to:

• Understand inequality notation and represent solutions on a number line.
• Solve linear inequalities and learn the crucial sign change that occurs when multiplying or dividing by a negative number.
• Formulate inequalities from worded constraints.
• Solve a linear inequality and depict its solution graphically.
• Properly reverse the inequality sign after performing operations involving negative numbers.

What is an Inequality?

An inequality is a mathematical statement that describes the relationship between two expressions that are not necessarily equal. Instead of equality, inequalities can express conditions where one quantity is greater than, less than, or equivalent to another quantity. The most common symbols used in inequality notation are:

• $ < $: Less than
• $ > $: Greater than
• $ \leq $: Less than or equal to
• $ \geq $: Greater than or equal to

For example, the inequality $ x < 5 $ means that $ x $ can take any value less than 5. In contrast, $ x \geq 3 $ means $ x $ can be any value that is 3 or greater, including 3 itself.

Representing Inequalities on a Number Line

To represent solutions to inequalities visually, we can use a number line. Let's take the inequality $ x < 3 $ as an example. Below are the steps to represent it:

1. Draw a number line: A horizontal line with numbers marked evenly.
2. Locate the number 3: Mark 3 on the number line.
3. Choose an appropriate symbol: Since the inequality is $ < $, we will draw an open circle at 3 to indicate that 3 is not included in the solution. If the inequality were $ \leq $, we would use a closed circle.
4. Shade the appropriate region: Shade the line to the left of 3 to denote all values less than 3.

Example 1

Problem: Represent the solution to the inequality $ x \geq -2 $ on a number line.

1. Draw a number line.
2. Mark the number -2 on it.
3. Use a closed circle on -2 since it’s included in the solution (due to $ \geq $).
4. Shade the line to the right of -2, indicating all numbers greater than or equal to -2.

Solving Linear Inequalities

Solving linear inequalities is similar to solving linear equations, but with one key difference: we must keep track of the inequality sign. This can change depending on the operation performed, specifically when multiplying or dividing both sides of the inequality by a negative number.

Key Rule: Sign Change

When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. For example, if we have:

$$

-2x < 6

$$

Dividing by -2 gives:

$$

$\frac{-2x}{-2}$ > $\frac{6}{-2}$ \quad \Rightarrow \quad x > -3

$$

Here we changed the inequality from $ < $ to $ > $.

Example 2

Problem: Solve the inequality $ 3x - 4 \leq 5 $.

Step 1: Add 4 to both sides:

$$

3x - 4 + $4 \leq 5$ + 4

$$

$\Rightarrow 3x \leq 9$

Step 2: Divide both sides by 3:

$$

$\frac{3x}{3}$ $\leq$ $\frac{9}{3}$ \quad \Rightarrow \quad x $\leq 3$

$$

Graphing the Solution

To graph the solution $ x \leq 3 $, you would:

1. Mark 3 with a closed circle on the number line.
2. Shade to the left, showing all values less than or equal to 3.

Forming Inequalities from Word Problems

Often, you will encounter real-world situations that can be modeled with inequalities. The first step is to translate the words into an inequality.

Example 3

Problem: A store sells t-shirts for $ \$15 $ each. You have a budget of $ \$60 $. Write and solve an inequality representing the number of t-shirts $ x $ you can buy.

Step 1: Set up the inequality for your budget:

$$

$15x \leq 60$

$$

Step 2: Divide both sides by 15:

$$

$\frac{15x}{15}$ $\leq$ $\frac{60}{15}$ \quad \Rightarrow \quad x $\leq 4$

$$

This means you can buy a maximum of 4 t-shirts. To represent this on a number line, mark 4 with a closed circle and shade to the left.

Reversal of Inequality Sign

It is essential to remember the sign change rule when manipulating inequalities. This rule is crucial when expressions involve negatives.

Example 4

Problem: Solve the inequality $ -4x > 12 $.

Step 1: Divide by -4 (remember to reverse the inequality):

$$

$\frac{-4x}{-4}$ < $\frac{12}{-4}$ \quad \Rightarrow \quad x < -3

$$

Conclusion

In this lesson, we have covered the basics of linear inequalities, how to represent them on a number line, and the importance of reversing the inequality sign when multiplying or dividing by negative numbers. You should be comfortable solving inequalities and translating real-world situations into mathematical terms. As you practice, remember to stay vigilant about these signs and processes.

Study Notes

• Inequality Notation: Learn the symbols $ <, >, \leq, \geq $.
• Graphing: Open and closed circles represent whether endpoints are included.
• Key Rule: Reverse the inequality sign when multiplying or dividing by a negative number.
• Applications: Formulate inequalities from scenarios and solve them.
• Practice: Regularly work through examples to solidify your understanding.