Problem 35 - Algebra 1: Concepts and Skills

Question

Check to see if the given value of the variable is or is not a solution of the inequality. $$ 42+x \leq 65 ; x=23 $$

Step-by-Step Solution

Answer

Yes, x=23 is a solution to the inequality.

Step 1: Substitute variable

Substitute the given value of x which is 23 into the inequality. Thus the inequality becomes: \(42 + 23 \leq 65\)

Step 2: Perform the Addition

Now, add the numbers, resulting in \(65 \leq 65\)

Step 3: Compare the Numbers

As observed, 65 is indeed equal to 65. Since the inequality allows for the two sides to be equal (as indicated by the \(\leq\) symbol), the inequality is true.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that determine the relative size of two values, indicating whether one is less than, greater than, or equal to the other. They use symbols such as ">", "<", and "≤". The expression "\(a \leq b\)" means that \(a\) is less than or equal to \(b\). These expressions are crucial in math because they help set boundaries or restrictions in problems.

When solving inequalities, the goal is to determine the range of values that satisfy the condition. For instance, if we have \(42 + x \leq 65\), we are looking for all values of \(x\) that make the statement true. This involves isolating the variable and finding its potential values.

The Role of Substitution in Solving Inequalities

Substitution is a method used in algebra to simplify expressions or equations by replacing variables with actual numeric values. This technique can help quickly verify if a specific number solves an inequality. In our given problem, substituting the value 23 for \(x\) in \(42 + x \leq 65\) simplifies the inequality to \(42 + 23 \leq 65\).

The simplicity of substitution lies in its straightforwardness—it transforms the problem into a basic arithmetic operation which is easier to handle. This helps in checking whether the chosen value lies within the solution set of the inequality. Substitution reduces the cognitive load, focusing only on numbers after the replacement.

Comparison of Numbers in Inequalities

Once variables are substituted and arithmetic is performed, the next step involves comparing the numbers to determine if the inequality holds true. In the example \(65 \leq 65\), it is clear that 65 is indeed equal to 65.

With inequalities involving the symbol "\(\leq\)", the condition is satisfied if the left-hand side is less than or exactly equal to the right-hand side. This step reassures us that we have found the correct range or specific value that makes the inequality true.

• This step eliminates doubts and confirms whether the initial substitution was appropriate.
• Comparison solidifies the solution by clarifying that all operations leading to this result were correct.

Understanding how to compare numbers effectively is key to mastering inequalities and ensuring accuracy in mathematical reasoning.