Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$a-7 \geq 15 ; 22$$

Short answer

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

Step-by-step solution

Substitute the number into the inequality

Replace the variable \(a\) with the given number (22) in the inequality \(a-7 \geq 15\). Which gives us \(22-7 \geq 15\)

Perform the calculation

Calculate the left side of the inequality \(22-7\), which results in 15. Which gives us \(15 \geq 15\)

Evaluate the inequality

The result \(15 \geq 15\) is a true statement, since 15 is indeed greater than or equal to 15.

Key concepts

Checking Solutions of Inequalities

When approaching an inequality, such as \( a-7 \geq 15 \), you're presented with a claim that for certain numbers, called solutions, the inequality holds true. To 'check solutions of inequalities,' you essentially prove or disprove that these numbers satisfy the inequality.

To do this effectively, you must understand what an inequality is. An inequality is like an equation, but instead of saying two things are exactly the same, it states that one quantity is either larger or smaller than another.

Here's how to check a solution:

• Identify the inequality and the proposed solution. In our example, the inequality is \( a-7 \geq 15 \), and the proposed solution is 22.
• Substitute the solution into the inequality. Replace the variable with the proposed solution value. The resulting statement will either be true or false.
• Evaluate the truth of the statement. If, after substitution, the statement makes sense (like 15 is indeed greater than or equal to 15), then the proposed number is a solution to the inequality.

Substitute Values into Inequalities

The process of 'substitute values into inequalities' is pivotal in determining the truth of the inequality for a particular value. Just like plugging numbers into equations, replacement provides a hands-on method to verify solutions.

To substitute correctly, follow these steps:

• Write down the inequality. Make sure you have clear space where the variable is placed.
• Insert the value in place of the variable. In the original inequality \( a-7 \geq 15 \), 'a' is our variable. If we are checking the number 22, we substitute 'a' for 22.
• Simplify the inequality. Perform the necessary arithmetic using the substituent. Here, we calculate \( 22-7 \) to obtain 15.

Substitution is an essential tool and must be done carefully to prevent errors in the solution checking process.

Evaluate Inequalities

When faced with the final step to 'evaluate inequalities,' we sum up the previous work and judge whether we indeed have a true statement or a false one.

The key here is to look at your simplified mathematical expression and compare the terms according to the inequality symbol used. With our given inequality \( a-7 \geq 15 \), after substitution and simplification, we obtained \( 15 \geq 15 \).

Here are things to consider:

• Understand the symbols. Know that '\(\geq\)' means 'greater than or equal to' and '\(\leq\)' means 'less than or equal to.' For '\( > \)' and '\( < \)', these are strict inequalities without the 'or equal to' part.
• Compare the values. If the statement reads \( 15 \geq 15 \), it's clear that 15 is equal to 15, so the initial inequality holds true for the value substituted.
• Make your conclusion. State whether your evaluated statement is true or false. In this scenario, the evaluation is true which confirms that the inequality is valid for the substituted value.

Remember, evaluating an inequality is essentially comparing numeric values to decide on the logical correctness of the proposed relationship.