Lesson 3.4: Inequalities and Absolute Value

Introduction

In this lesson, we will explore the concepts of inequalities and absolute value, two critical elements of algebra necessary for successful quantitative reasoning on the GMAT.

Objectives

Understand how to solve inequalities and reason with them.
Learn the behavior of signs when multiplying or dividing by negatives.
Define absolute value in terms of distance, and identify its various cases.
Solve inequality and absolute-value problems correctly.
Effectively handle sign-flip and case-split situations without making errors.

Hook

Imagine you are trying to find out how much money you need to save monthly to afford a new car, and you have a budget constraint that keeps changing. How would you express and solve such a problem using inequalities? In this lesson, we will develop the tools to do just that, ensuring you can tackle similar questions with ease.

1. Solving Inequalities

Understanding Inequalities

An inequality is a mathematical statement that compares two expressions and shows a relation between them using symbols such as $ < $, $ > $, $ \leq $, or $ \geq $. Unlike equations, which represent equality, inequalities indicate a range of possible values.

Example 1: Basic Inequality

Consider the inequality:

$$2x + 3 < 11$$

To solve for $ x $, follow these steps:

Subtract 3 from both sides:

$$2x < 11 - 3$$

$$2x < 8$$

Divide both sides by 2:

$$x < 4$$

Thus, the solution indicates that any number less than 4 satisfies the inequality. This can be represented on a number line by shading everything to the left of 4.

Sign Behavior with Inequalities

When solving inequalities, it is crucial to understand how the direction of the inequality changes when multiplying or dividing both sides by a negative number.

Example 2: Flipping the Inequality Sign

Consider the inequality:

$$-3x > 9$$

To isolate $ x $, we must divide both sides by -3. Remember, dividing by a negative flips the inequality sign:

Dividing by -3:

$$x < -3$$

The final solution tells us that $ x $ must be less than -3. If you were to test numbers greater than -3, they would not satisfy the original inequality, confirming our solution.

2. Absolute Value as Distance

Understanding Absolute Value

Absolute value measures the distance of a number from 0 on the number line, disregarding the direction. The absolute value of $ x $ is denoted as $ |x| $. This means:

If $ x \geq 0 $, then $ |x| = x $
If $ x < 0 $, then $ |x| = -x $

Example 3: Evaluating Absolute Values

Evaluate $ |5| $ and $ |-5| $:

$ |5| = 5 $
$ |-5| = 5 $

Thus, both values are the same distance from 0 on the number line, which helps solidify the concept of absolute value.

3. Solving Absolute Value Inequalities

Solving inequalities involving absolute values often requires breaking them into cases based on the definition of absolute value.

Example 4: Absolute Value Inequality

Consider the inequality:

$$|x - 2| < 5$$

To solve this, we need to identify the two cases:

Case 1: When $ x - 2 \geq 0 $ $\Rightarrow x - 2 < 5$ leads to

$$x < 7$$

Case 2: When $ x - 2 < 0 $ $\Rightarrow -(x - 2) < 5$ leads to

$$-x + 2 < 5$$

$$-x < 3$$

$$x > -3$$

Combining these two results, we find:

$$-3 < x < 7$$

This interval gives us all the $ x $ values that satisfy the original inequality.

4. Common Misconceptions

When working with inequalities and absolute values, students frequently make the following mistakes:

Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
Misapplying the definition of absolute value, leading to incorrect case splits.
Not properly representing final answers in inequality notation or as interval notation.

Conclusion

In this lesson, we have examined inequalities and absolute value, focusing on their definitions, solving methods, and applications. By understanding how to handle sign behavior and absolute value cases, you are now better equipped to tackle GMAT quantitative reasoning problems effectively.

Study Notes

Inequalities use $ < $, $ > $, $ \leq $, and $ \geq $.
Remember to flip the inequality sign when multiplying or dividing by a negative.
Absolute value $ |x| $ represents the distance of $ x $ from 0.
Break absolute value inequalities into cases to solve.
Check your work to avoid common misconceptions.