Problem 52 - 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. $$ a-7 \geq 15 ; a=22 $$

Step-by-Step Solution

Answer

Yes, \(a = 22\) is a solution to the inequality \(a - 7 \geq 15\).

Step 1: Substitute the value of 'a'

Substitute the given value, \(a = 22\), in place of \(a\) in the inequality \(a - 7 \geq 15\), giving \(22 - 7 \geq 15\).

Step 2: Calculation

Now simplify \(22 - 7\) to get \(15\), which gives us the new inequality \(15 \geq 15\).

Step 3: Analyze the result

We have ended up with the statement \(15 \geq 15\), which is true as 15 is indeed greater than or equal to 15. Therefore, \(a = 22\) is a solution to the inequality \(a - 7 \geq 15\).

Key Concepts

Solving Inequalities

Inequalities are similar to equations but instead of an equal sign, they use symbols like \( >, <, \geq, \text{and} \leq \). Solving inequalities means finding all possible values of a variable that make the inequality true. For instance, in the inequality \( a - 7 \geq 15 \), we are looking for values of \( a \) such that when 7 is subtracted from them, the result is 15 or more. Unlike equations, inequalities can have a range of solutions.
To solve this specific inequality, we isolate \( a \) by reversing the operations: we start by adding 7 to both sides of the inequality to eliminate the subtraction, resulting in \( a \geq 22 \). This tells us that any value of \( a \) which is 22 or greater will satisfy the inequality. Therefore, solving inequalities involves finding this range or set of solutions that match the conditions set by the inequality symbol.

Checking Solutions

Once we have a potential solution or set of solutions for an inequality, the next step is to check if a specific value truly satisfies the inequality. This process ensures that each solution is correct and meets the conditions specified by the inequality.
When checking if \( a = 22 \) is a solution to \( a-7 \geq 15 \), we substitute and simplify the expression as done in the previous steps. If our simplified expression follows the inequality condition (like \( 15 \geq 15 \)), then the given value is indeed a solution.
This verification process is crucial, especially in cases involving multiple potential solutions. It helps confirm whether each proposed solution holds true when plugged back into the original inequality.

Substituting Values

Substitution is the act of replacing a variable with a given number. It allows us to check specific values against inequalities or equations. Let's illustrate with the inequality \( a-7 \geq 15 \). To check if \( a = 22 \) is a solution, we substitute 22 into the place of \( a \). This converts the inequality into \( 22 - 7 \geq 15 \).
The next step is a simple arithmetic calculation to see if the resulting statement is true or false. In this instance, \( 22 - 7 \) simplifies to 15, leading to the statement \( 15 \geq 15 \). Since this statement is true, substituting \( a = 22 \) indeed satisfies the inequality.

• Substitution helps verify if given values are true solutions.
• It is critical in testing the suitability of specific values in inequalities.
• It simplifies expressions for better evaluation.

Substituting values is an essential skill in algebra that aids in validating potential solutions efficiently.