Problem 9 - Elementary Algebra

Question

Determine whether the given number is a solution to the given inequality. $$ 2 x+1<-3 \text { or } 2 x+1 \geq 5 ; x=2 $$

Step-by-Step Solution

Answer

Yes, \(x=2\) is a solution to the inequality.

Step 1: Understand the Inequality

The problem presents an inequality with two parts: \(2x+1 < -3\) or \(2x+1 \geq 5\). We need to check if \(x=2\) satisfies either part of this inequality.

Step 2: Substitute \(x=2\) into the First Inequality

Substitute \(x=2\) into the first part of the inequality:\[2(2) + 1 < -3\]This simplifies to:\[4 + 1 < -3\]\[5 < -3\]This statement is false, so \(x=2\) does not satisfy the first inequality.

Step 3: Substitute \(x=2\) into the Second Inequality

Next, substitute \(x=2\) into the second part of the inequality:\[2(2) + 1 \geq 5\]This simplifies to:\[4 + 1 \geq 5\]\[5 \geq 5\]This statement is true, so \(x=2\) satisfies the second inequality.

Key Concepts

Substitute Variable

In mathematical inequalities, substituting a variable means replacing the variable with a concrete number to determine if the inequality holds. In this exercise, our task is to replace the variable \(x\) with the given value, \(x = 2\). This substitution allows us to transform the abstract inequality into a simple numerical comparison.To perform the substitution, we plug \(x = 2\) into both parts of the compound inequality:

• First Inequality: \(2x + 1 < -3\)
• Second Inequality: \(2x + 1 \geq 5\)

By substituting \(x = 2\), both inequalities can be directly evaluated to understand if \(x = 2\) is indeed a solution.

Inequality Evaluation

Once we substitute the variable, we proceed with evaluating each resulting statement to see if it is true or false. ### Evaluating the First InequalityFor the inequality \(2x + 1 < -3\), substituting \(x = 2\) yields:\[2(2) + 1 < -3 \4 + 1 < -3 \5 < -3\]In this case, 5 is not less than -3, indicating the statement is false.### Evaluating the Second InequalityFor the second part \(2x + 1 \geq 5\), substituting \(x = 2\) results in:\[2(2) + 1 \geq 5 \4 + 1 \geq 5 \5 \geq 5\]Here, the left side is equal to the right side, and since 5 is equal to 5, the statement is true. Thus, \(x = 2\) satisfies the second inequality. Through evaluation, we confirm that despite the falsehood in the first inequality, \(x = 2\) still works due to its truth in the second.

Step by Step Solution

The process of solving inequalities is much like a step-by-step recipe. Each step builds on the other to eventually reveal whether a solution satisfies the problem. Here's a concise walkthrough:1. **Understand the Inequality Context**: Start by breaking down the problem into its components. Recognize it as a compound inequality with an "or" joining the two parts.
2. **Substitute the Variable**: Use the given value for \(x\) in each inequality separately, transforming the statement from a variable-based to a numerical one.
3. **Evaluate Each Part**: Once substituted, check each inequality. If at least one part holds true, as seen in the second inequality, then \(x = 2\) is a solution to the compound inequality.
By following these steps methodically, you can effectively determine the truth of inequalities. This approach emphasizes understanding each part and ensures no critical steps are overlooked, setting up a solid mathematical foundation for solving inequality problems.