Problem 75 - Algebra 1

Question

Check whether the given number is a solution of the inequality. $$ 4 x \leq 28 ; 7 $$

Step-by-Step Solution

Answer

Yes, the number 7 is a solution to the inequality \(4x \leq 28\)

Step 1: Identify the given number and the inequality

The inequality to test is \(4x \leq 28\), and the number to check in the inequality is 7.

Step 2: Substitute the given number in the inequality

Replace \(x\) in the inequality with 7 to check the validity. It results in \(4 * 7 \leq 28\)

Step 3: Simplify the inequality

Doing the multiplication, we get \(28 \leq 28\)

Step 4: Check the validity of the simplified inequality

Since 28 is equal to 28, the inequality holds true.

Key Concepts

Substitution in Inequalities

When you have an inequality like \( 4x \leq 28 \) and a number you want to check, substitution is the method to use. Here, you're asked to see if the number 7 makes this inequality true. The process involves replacing the variable, which is \( x \), with the given number, which in this case is 7. By doing this, you test if the number satisfies the inequality condition. For example, substitute by performing:

• Replace \( x \) with 7, giving you \( 4 \times 7 \leq 28 \).

Substitution is a straightforward operation but crucial for checking solutions in inequalities.

Steps to Solve Inequalities

Solving inequalities is akin to solving equations, but with a few additional steps to consider.

• First, identify the inequality and what you need to check. For instance, the inequality is \(4x \leq 28\), and our check is with the number 7.
• Next, substitute the provided number for the variable, converting the inequality into a statement you can calculate. For this example, it transforms into \(4 \times 7 \leq 28\).
• Then, perform the necessary arithmetic operations. Calculating gives you \(28 \leq 28\).
• Finally, compare the results to check the truth of the inequality. Here, 28 is indeed less than or equal to 28, so the inequality is satisfied.

Follow these steps each time for a thorough understanding and correct solving of inequalities.

Validity of Inequalities

Determining the validity of an inequality is the key to understanding if a solution is correct. Validity means checking whether your final expression upholds the inequality's condition. The noteworthy point is:

• After substitution and simplification, if the inequality holds true — like \(28 \leq 28\) in our case — the test number is a valid solution to the inequality.
• Remember that "less than or equal to" (\(\leq\)) includes both cases where the first number is either less than or exactly equal to the second number.

Thus, knowing how to check this correctly ensures that you can reliably determine when an inequality holds true.