Problem 34 - 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. $$ 14-y>12 ; y=4 $$

Step-by-Step Solution

Answer

The value \(y = 4\) is not a solution of the inequality \(14-y > 12\).

Step 1: Substitute the given value of y

First of all, substitute the given value of y into the inequality. The inequality is \(14-y > 12\), so replace y with 4. You will get \(14 - 4 > 12\), which simplifies to \(10 > 12\).

Step 2: Compare the result

Now compare the obtained result with the original inequality. Actually, \(10 > 12\) is not true, as 10 is not greater than 12.

Step 3: Conclusion

Since the inequality \(10 > 12\) is not true, it implies that \(y = 4\) is not a solution to the original inequality \(14-y > 12\). Therefore, 4 is not a solution of the inequality.

Key Concepts

Substitution Method

When solving inequalities, one common technique is the substitution method. This involves replacing a variable with a given value to determine if it satisfies the inequality. For example, in the inequality \(14-y > 12\), if we are given \(y = 4\), we substitute 4 in place of \(y\). The expression then becomes \(14 - 4 > 12\). This simplifies the inequality to \(10 > 12\).

To apply the substitution method, follow these steps:

• Identify the variable: Look for the variable you need to substitute in the inequality.
• Replace with given value: Swap the variable with the value provided.
• Simplify: Perform the arithmetic operations to simplify the inequality.

By comparing the resulting expression, you can determine if the inequality holds true, thus identifying whether the substituted value is a valid solution.

Inequalities

Inequalities are mathematical expressions that indicate a relationship between two values where they are not equal. Inequalities use symbols like \(>\), \(<\), \(\geq\), and \(\leq\). These symbols help describe if one quantity is greater than, less than, or possibly equal to another.

Understanding inequalities involves knowing:

• Symbols: Recognize the different inequality signs and their meanings.
• Solving process: Manipulate the inequality to maintain true statements (e.g., when adding or subtracting numbers from both sides, the inequality remains unchanged).
• Comparison: After simplifying, check whether the values satisfy the initial inequality condition.

In our example, the inequality \(14-y > 12\) means the expression on the left, after substituting \(y\), should be greater than 12. However, after substitution, \(10 > 12\) turns false, so the inequality does not hold. This shows that the given value does not satisfy the inequality.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition and subtraction). They form the backbone of equations and inequalities, providing a way to describe mathematical relationships. In the context of our exercise, \(14-y\) is an algebraic expression.

When dealing with algebraic expressions:

• Identify terms: Each part of the expression separated by \(+\) or \(-\) is a term (e.g., in \(14-y\), \(14\) and \(-y\) are terms).
• Understand operations: Recognize how operations affect terms and how to manipulate them.
• Substitution: Replace variables with numbers to simplify and evaluate the expression.

Algebraic expressions are flexible. They can model real-world situations and solve problems when used correctly. By getting comfortable with expressions, you can better understand relationships within inequalities and find solutions.