Problem 18 - Algebra 1: Concepts and Skills

Question

Check to see if \(b=8\) is or is not a solution of the inequality. $$ 14-b \leq 3 $$

Step-by-Step Solution

Answer

\(\ b = 8\) is not a solution for the given inequality.

Step 1: Substitute the given value

Substitute the value of \(b=8\) into the inequality \(14-b \leq 3\). By substituting \(b=8\), we get \(14-8 \leq 3\).

Step 2: Simplify after substitution

Simplifying this further, we get \(6 \leq 3\).

Step 3: Evaluate the inequality

The inequality \(6 \leq 3\) is false because 6 is not less than or equal to 3.

Key Concepts

Inequality Evaluation

Inequality evaluation is the process of determining whether the given condition represented by an inequality holds true or false.
Understanding this concept is crucial when working with inequalities, as it enables us to check the validity of potential solutions.

In the context of our exercise, we need to check if substituting a specific value for a variable satisfies the inequality. For example, with the inequality \(14-b \leq 3\), we evaluate:

• Substitute the variable with the provided value, which leads us to check the new inequality \(14-8 \leq 3\).
• Simplify the expression and evaluate the truth of the inequality.
• In this specific case, simplifying \(14-8\) gives us 6, and 6 is not less than or equal to 3.

This evaluation shows the inequality is false with the substituted value, hence, \(b=8\) is not a solution.

Variable Substitution

Variable substitution is a fundamental technique in algebra that involves replacing a variable with a given number to simplify expressions or solve equations and inequalities.
Instead of dealing with an abstract symbol, substitution allows us to work with concrete numbers. This makes the problem easier to handle.

For our inequality \(14-b \leq 3\), variable substitution plays a critical role:

• The variable \(b\) is replaced by 8. This turns the inequality into \(14-8 \leq 3\).
• By performing this substitution, we break down the problem into simpler numeric forms that are more straightforward to evaluate.

Substitution not only simplifies our task but also provides a systematic way to plug in potential solutions and directly test them.

Algebraic Inequalities

Algebraic inequalities are expressions involving algebraic terms that use inequality symbols like \(<, \leq, >, \geq\).
These inequalities represent a range of values rather than a single number, unlike simple equations.

The inequality \(14-b \leq 3\) in our exercise is an example of how we use inequalities in algebra to compare expressions:

• The inequality signifies that the expression on the left (\(14-b\)) should be less than or equal to 3.
• Using algebraic methods, such as substitution, we determine whether this is true for a specific value of the variable.

Understanding algebraic inequalities is crucial as they form the basis for many math problems involving real-life scenarios, like budgeting, where constraints are represented by inequalities.