Problem 53 - Algebra 1: Concepts and Skills

Question

CHECKING SOLUTIONS OF INEQUALITIES Check to see if the given value of the variable is or is not a solution of the inequality. $$ 6+y \leq 8 ; y=3 $$

Step-by-Step Solution

Answer

No, \( y = 3 \) is not a solution of the inequality \( 6+y \leq 8 \).

Step 1: Identify the inequality and the given value of the variable

In this case, the inequality is \( 6+y \leq 8 \) and the given value for \( y \) is 3.

Step 2: Substitute the value of the variable into the inequality

Substitute \( y = 3 \) into the inequality: \( 6+3 \leq 8 \).

Step 3: Simplify the inequality

After performing the addition, we obtain: \( 9 \leq 8 \)

Step 4: Check if the inequality is true

The inequality \( 9 \leq 8 \) is not true. So, the provided value for \( y \) is not a solution of the given inequality.

Key Concepts

Checking Solutions in Inequalities

When dealing with inequalities, a common task is to determine if a specific value of the variable makes the inequality true or false. Let's consider the inequality \(6 + y \leq 8\) with \(y = 3\). Checking solutions means substituting the given value into the inequality and verifying if it holds.
Start by substituting \(y\) with the provided value to see if the inequality remains valid. Once this is done, you can easily check if the given value is a solution. If the inequality remains true after substitution, the value is a valid solution. Otherwise, it's not. This process helps confirm that your answers are indeed solving the problem as asked in the exercise.

Understanding the Substitution Method

The substitution method is a straightforward approach in solving inequalities or equations. It involves replacing a variable with a given number to simplify the inequality, making it easier to resolve.
In the exercise, we substitute \(y = 3\) into the inequality \(6 + y \leq 8\). This replacement helps change the inequality into a simple numerical expression, simplifying the process.

• Identify what number to replace the variable with.
• Carefully substitute the number for the variable in the inequality.
• Simplify the resulting expression.

By using substitution, you can more easily manage inequalities and identify whether a given value is a solution.

Simplifying Inequalities with Ease

Simplifying inequalities is an essential step in verifying solutions. After substitution, the inequality converts into a simpler form. This simplification process allows you to assess the accuracy of your result.
For the inequality \(6 + 3 \leq 8\), simplify the left side by performing the arithmetic: add 6 and 3 to get 9. Now, you have an inequality that reads \(9 \leq 8\). Recognizing that 9 is not less than or equal to 8, the inequality does not hold.
Simplifying involves:

• Performing basic operations like addition or subtraction.
• Comparing the result to the right side of the inequality.

This step is crucial as it reveals the truthfulness of the inequality after making the substitution.

Applying Mathematical Reasoning

Mathematical reasoning allows you to logically analyze the problem and the solution you've concluded. This involves understanding and interpreting the mathematical facts present in the inequality.
Once you've simplified \(9 \leq 8\), it is essential to reason through mathematical logic that 9 is indeed not less than or equal to 8. Hence, the initial value for \(y\) provided does not satisfy the inequality.
By engaging in mathematical reasoning, you ensure that you:

• Evaluate the logic behind each step of simplification.
• Confirm or refute whether the inequality solution claims are valid.
• Ensure that each step of your solution process is justifiable.

By practicing sound mathematical reasoning, you'll develop a stronger understanding of inequalities and their solutions.