Problem 22 - Algebra 1: Concepts and Skills

Question

Check to see if \(b=8\) is or is not a solution of the inequality. $$ 16 \leq b^{2} $$

Step-by-Step Solution

Answer

Yes, \(b = 8\) is a solution to the given inequality.

Step 1: Substitute the Value of b

Substitute \(b = 8\) into the inequality, which results in \(16 \leq 8^{2}\).

Step 2: Simplify the Inequality

Simplify the equation to get \(16 \leq 64\).

Step 3: Check If the Statement is True or False

The inequality \(16 \leq 64\) is a true statement, meaning that \(b = 8\) is a solution to the original inequality.

Key Concepts

Substitution Method

When solving inequalities, one of the methods we often use is the **substitution method**. This method involves replacing a variable with a chosen value to determine if it satisfies the inequality. In the exercise, the variable is \( b \), and the value we're testing is \( b=8 \). We substitute \( 8 \) for \( b \) in the inequality \( 16 \leq b^2 \).

Here's how to perform substitution:

• Identify the variable in the expression.
• Take the given value that you want to test — in this case, \( 8 \).
• Replace every occurrence of the variable with this value. For our problem, this means calculating \( 8^2 \).

After substitution, we find \( 16 \leq 8^2 \) which simplifies to \( 16 \leq 64 \). This step is crucial because it sets the stage for simplification and verification.

Inequality Simplification

Once substitution is carried out, the next step is **inequality simplification**. Simplification helps in identifying whether the inequality holds true after plugging in the variable's value. Post substitution in our example, the inequality becomes \( 16 \leq 64 \).

To simplify an inequality simplification generally involves:

• Evaluating powers or products in the inequality, like \( 8^2 = 64 \).
• Checking if the simplified form of the inequality makes sense by comparing the two sides.

Simplification not only clarifies the relationship between the numbers but also significantly aids in understanding whether the chosen value is a valid solution. Here, because \( 16 \) is clearly less than or equal to \( 64 \), the inequality holds true. This implies that the candidate value indeed satisfies the inequality.

Mathematical Reasoning

The application of **mathematical reasoning** is pivotal in moving beyond mere computation to validating results. In solving inequalities through the substitution method, logical reasoning helps confirm the truthfulness of the derived statement.

In the exercise, after substituting \( b=8 \), we computed \( 16 \leq 64 \) during simplification. Now, our task is to use reasoning to assess whether the inequality holds. This means asking: "Is \( 16 \) less than or equal to \( 64 \)?" The answer is obviously yes.

Engaging in such reasoning enhances problem-solving skills and provides a foundation for understanding more complex inequalities. It helps us ascertain if our calculations match the initial condition laid out by the inequality. Given the true statement, \( b=8 \) is indeed a solution to the inequality, confirming our reasoning process.