Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each inequality is true or false. $$-5 \leq-8$$

Step-by-Step Solution

Answer

The inequality \(-5 \leq -8\) is false.

Step 1: Identify Inequality

First, look at the inequality presented: \(-5 \leq -8\). The inequality is asserting that -5 is either less than or equal to -8.

Step 2: Evaluate Inequality

Next, compare the two numbers -5 and -8. Looking at the number line, we'll find that -5 is to the right of -8, meaning -5 is more than -8.

Step 3: Determine if the Given Inequality is True or False

Since we found that -5 is more than -8, not less than or equal to -8, we can conclude that the inequality is false.

Key Concepts

Number Line

A number line is a visual representation of numbers on a straight line. It helps us understand the relative order of numbers and how they compare. Imagine a straight line where numbers increase as you move to the right and decrease as you move to the left.
Negative numbers, such as -5 and -8, are on the left side of zero on the number line. Since -8 is further left on the number line than -5, it means -8 is smaller than -5.
Understanding this layout helps tremendously when comparing numbers or deciding the truth value of inequalities.

• Numbers further right are greater.
• Numbers further left are lesser.

Using a number line can make solving inequalities easier by providing a visual way to compare numbers.

Comparison of Numbers

Comparing numbers is an essential skill in understanding mathematical inequalities. It involves determining if one number is greater than, less than, or equal to another. This is often done using symbols like < (less than), > (greater than), and = (equal to).
For the inequality \(-5 \leq -8\), we are asked if -5 is less than or equal to -8. To compare these numbers, we look at their positions on a number line. Since -5 is located to the right of -8, -5 is actually greater than -8.
If you remember that on the number line, greater numbers are always to the right, it makes comparison clearer. This understanding allows us to quickly determine the truth of inequalities by visualizing or using simple arithmetic concepts.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators like addition, subtraction, multiplication, and division. They form the core elements when working with inequalities, especially when variables come into play.
For inequalities like \(-5 \leq -8\), we don't have variables, but it's crucial to grasp the inequality symbols and their meaning. When you encounter algebraic expressions that include variables, you will follow similar steps to evaluate their correctness.

• Understand each part of the expression.
• Evaluate by substituting known values if possible.
• Use the properties of inequalities to guide you.

Mastering algebraic expressions will also support your ability to solve more complex inequalities where variables need to be isolated or solved for.