Problem 55 - Algebra 1

Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$n(21-n)<100 ; 8$$

Step-by-Step Solution

Answer

No, 8 does not satisfy the inequality \(n(21-n)<100\).

Step 1: Substitute the given number into the inequality

First, substitute 8 into the inequality replacing n: \(8(21 - 8)\).

Step 2: Simplify the inequality

Next, simplify the multiplication: \(8 * 13 = 104\). This gives \(104 < 100\).

Step 3: Evaluate the inequality

Determine whether \(104 < 100\) is true. In this case, since 104 is not less than 100, the inequality is false.

Key Concepts

Inequality Substitution

Inequality substitution is a method to determine if a particular value is a solution to an inequality. It involves replacing the variable in the inequality with the given value and checking to see if the resulting statement is true.

For instance, the given exercise requires checking if 8 is a solution to the inequality \(n(21-n) < 100\). By substituting 8 in place of \(n\), we obtain \(8(21-8)\), which simplifies to \(104\). But, the inequality \(104 < 100\) is not true, indicating that 8 is not a solution to the original inequality.

It is crucial to carefully substitute the given value everywhere the variable appears in the inequality to avoid errors and to obtain an accurate simplification of the inequality for further evaluation.

Simplifying Inequalities

Simplifying inequalities is a necessary step after substituting the given number for the variable. It often involves arithmetic operations such as addition, subtraction, multiplication, or division. The goal is to obtain a simpler or more obvious form of the inequality to make the evaluation process straightforward.

In our exercise, after the substitution step gives us \(8(21-8)\), the simplification involves multiplying the numbers together. Doing so reduces the inequality to \(104 < 100\). While simplification may sometimes lead to a clear immediate answer, at other times it serves as a prelude to further analysis or evaluation of the inequality.

Evaluating Inequalities

Evaluating inequalities is the final step to ascertain if the simplified mathematical statement holds true. This is where we scrutinize the simplified form after substitution and simplification, to conclude whether the original statement is satisfied or not.

In our example, the simplified form \(104 < 100\) needs to be evaluated. Quickly, we recognize that 104 is greater than 100, not less. Thus, the evaluation of this inequality shows us that the statement is false, telling us that the number 8 is not a solution to the inequality \(n(21-n)<100\).

Evaluating inequalities accurately helps in understanding the solution set of an inequality, which is critical in solving various real-world problems that can be modeled by inequalities.