Problem 20 - Algebra 1: Concepts and Skills

Question

Check to see if \(b=8\) is or is not a solution of the inequality. $$ 8 \geq 64 \div b $$

Step-by-Step Solution

Answer

Yes, \(b=8\) is a solution of the inequality \(8 \geq \frac{64}{b}\).

Step 1: Substitute \(b\) with 8

Replace \(b\) with 8 in the inequality, resulting in the expression: \(8 \geq \frac{64}{8}\)

Step 2: Simplify the Right Hand Side

Simplify the expression on the right side of the inequality: \(\frac{64}{8} = 8\) which gives us a new inequality: \(8 \geq 8\)

Step 3: Check the Inequality

Check whether the resulting inequality is true. Since 8 is equal to 8, the inequality \(8 \geq 8\) is true.

Key Concepts

Substitution

Substitution is a fundamental technique in solving inequalities, equations, and various mathematical problems. It involves replacing a variable with a given number or expression. This allows us to convert an abstract problem into a concrete one and see if a specific value satisfies an inequality or condition.

In the original exercise, we are tasked with determining if \( b=8 \) is a solution to the inequality \( 8 \geq \frac{64}{b} \). The first step is to substitute the variable \( b \) with the given number 8. This transforms the inequality into:

• \( 8 \geq \frac{64}{8} \)

This straightforward substitution lets us deal with numbers we understand, making the next steps of solving the problem easier.

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics that helps make equations and inequalities easier to understand and solve. It involves performing operations to reduce an expression to its simplest form.

After substitution in our exercise, the inequality becomes \( 8 \geq \frac{64}{8} \). Here, the next task is to simplify the right-hand side of the inequality:

• First, divide: \( \frac{64}{8} = 8 \)

With the expression simplified, the inequality now reads \( 8 \geq 8 \). This step highlights how simplifying allows us to directly compare both sides of the inequality.

Checking Solutions

Checking solutions is a crucial step in verifying the validity of an answer in any mathematical problem. Once we have substituted and simplified, we must ensure that our results make sense within the context of the problem.

In this exercise, after substituting \( b=8 \) and simplifying, we arrived at the inequality \( 8 \geq 8 \). To check this solution, consider:

• The left-hand side of the inequality is 8.
• The right-hand side, after simplification, is also 8.

Since the inequality \( 8 \geq 8 \) holds true (because 8 is indeed equal to 8), we conclude that \( b=8 \) is a valid solution to the original inequality. This comprehensive verification ensures the correctness of our solution, giving us confidence in our mathematical reasoning.