Problem 55 - Algebra 1

Question

Is the given number a solution of the inequality? $$8+5 x<-2 ;-2$$

Step-by-Step Solution

Answer

No, -2 is not a solution to the inequality \(8 + 5x < -2\).

Step 1: Substitute the value

Substitute the given value of \(x\) into the inequality. So, you have \(8 + 5(-2)\).

Step 2: Simplify the inequality

Simplify the inequality by performing the multiplication. This gives you \(8 - 10\).

Step 3: Calculate the sum

Calculate the sum, that should result in \(-2\).

Step 4: Verify the inequality

Now verify if \(-2\) is less than \(-2\). If it is, the number given is a solution to the inequality otherwise it's not.

Key Concepts

Substitution Method

The substitution method is a useful algebraic technique for solving inequalities and equations. It involves replacing a variable with a given number to see if it satisfies the inequality. In this exercise, we substitute \(x = -2\) into the inequality \(8 + 5x < -2\). This initial step sets the stage for verifying if the inequality holds true.

To perform the substitution, simply take the number provided, \(-2\), and plug it in place of \(x\) in the expression. This action transforms the inequality from \(8 + 5x < -2\) into \(8 + 5(-2) < -2\).

Using the substitution method allows you to determine if a specific value is a potential solution by checking if the statement is true. If, after simplification, the inequality holds, then the number is a solution; if not, then it is not a solution.

Simplification

Simplification involves reducing an expression or inequality to its simplest form. This step is vital in ensuring that the inequality’s truth can be easily assessed. In the current scenario, after substituting, we have \(8 + 5(-2) < -2\).

Begin by calculating the multiplication. Multiply \(5(-2)\), which equals \(-10\). This modifies the inequality to \(8 - 10 < -2\).

The next part of simplification is to perform the basic arithmetic operation of subtraction, so subtract \(10\) from \(8\), resulting in \(-2\). After these steps, the inequality becomes \-2 < -2\.

Simplifying an inequality reduces it to a straightforward expression where the truth or falsity can be directly evaluated.

Verification of Solutions

Verification is the stage where you confirm if the number substituted into the inequality indeed satisfies it.

Here, after simplification, we obtained \-2 < -2\. Now, evaluate if \-2\ is less than \-2\. According to the rules of inequalities, one number is only considered less than another if it is indeed smaller.

Since \-2\ is not less than \-2\ (as both are equal), \(-2\) does not satisfy the requirement of being less than \-2\.

Understanding the verification step is crucial, as it clearly showcases whether the tested value meets the inequality condition, reinforcing the accuracy of your solution.