Problem 52 - Algebra 1

Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$\frac{c+5}{3} \leq 4 ; 3$$

Step-by-Step Solution

Answer

Yes, the given number 3 is indeed a solution to the inequality \(\frac{c+5}{3} \leq 4\).

Step 1: Substitute the Value

Replace variable 'c' in the inequality equation \(\frac{c+5}{3} \leq 4\) with the given number value which in this case is 3. The equation now becomes: \(\frac{3+5}{3} \leq 4\)

Step 2: Simplify the Equation

Perform the sum operation in the numerator of the fraction. The equation will become \(\frac{8}{3} \leq 4\).

Step 3: Check the Inequality

\(\frac{8}{3}\) when converted into decimal gives approximately 2.67. Hence, 2.67 is less than 4. So, the inequality \(\frac{8}{3} \leq 4\) holds true.

Key Concepts

Checking Solutions

To determine if a number is a solution to an inequality, we need to check if it satisfies the inequality. In simple terms, for any given inequality, substituting the provided number into the inequality should result in a true statement. For example, if you have the inequality \(\frac{c+5}{3} \leq 4\) and the number you want to check is 3, you would replace \(c\) with 3 and see if the resulting statement is true.

In our case, substituting 3 into the inequality gives us \(\frac{3+5}{3} \leq 4\). If this statement holds true after simplifying, then 3 is indeed a solution to the inequality. Therefore, checking solutions involves three main steps: substituting the number, simplifying the equation, and verifying if the inequality is satisfied.

Substitution in Inequalities

Substitution is a fundamental technique used to solve inequalities. It's a process where we replace a variable with a given number to see if the inequality can be converted into a true statement.

Here’s how you do it step by step:

• Identify the variable in the inequality. In our example, it’s \(c\) in \(\frac{c+5}{3} \leq 4\).
• Note the number provided to check, in this instance, it's 3.
• Replace the variable with the given number. So, \(c\) becomes 3, turning the inequality into \(\frac{3+5}{3} \leq 4\).

After substitution, you’re left with a numerical inequality that is ready for the next step: simplification.

Inequality Simplification

Once substitution is complete, the next step is simplifying the inequality. Simplification involves performing basic arithmetic operations on both sides of the inequality to see if the statement holds true.

In our example, after substituting 3 for \(c\), you have \(\frac{3+5}{3} \leq 4\). Start by performing the operation in the numerator: 3 plus 5 equals 8. This changes the equation to \(\frac{8}{3} \leq 4\).

Next, simplify \(\frac{8}{3}\) to its decimal form, which is approximately 2.67. Then, you need to compare this decimal with 4. Since 2.67 is indeed less than 4, our initial inequality stands true. Simplification concludes with verifying the inequality's correctness, ensuring the substituted number is a viable solution.