91Ó°ÊÓ

To solve inequalities, much like solving equations, we aim to find the range of values a variable can take that makes the inequality true. When solving inequalities, we perform similar operations as equations, such as adding or subtracting the same value from both sides and multiplying or dividing both sides by the same number. However, there is a key rule to remember: when multiplying or dividing both sides by a negative number, the inequality sign must be flipped. This is a crucial distinction compared to solving equations. In the exercise given, our goal is to isolate the variable and determine if there are any values for which the inequality holds true. In this example, after simplifying, we discovered a statement that is false (3 is not less than -4), indicating there are no possible solutions for the inequality.

Let's dive into the step-by-step process of solving the given inequality: - **Identify the inequality**: Start with the inequality provided: \( x + 3 < x - 4 \).- **Eliminate the variable from one side**: Subtract \( x \) from both sides. This aims to simplify the inequality by removing \( x \) from one side. After doing this, we are left with: \( 3 < -4 \).These steps help to clear one side of the inequality from the variable, making it easier to evaluate.- **Evaluate the result**: The resulting statement, \( 3 < -4 \), is false. This false statement indicates the absence of any real number \( x \) that can satisfy the original inequality. By going through these steps, we see that solving inequalities is about carefully performing operations that simplify the inequality without changing its meaning.

Once inequalities are solved, it’s essential to interpret the results correctly. In some cases, like the one in our exercise, the simplification may lead to a statement that doesn’t hold true, such as \( 3 < -4 \). Here, it's important to recognize that - **No Solution Exists**: If the inequality transforms into a false statement, it indicates there are no values for the variable that will satisfy it. - **All Values Invalid**: The impossibility of any solution suggests that there is no range of \( x \) values meeting the inequality condition.Understanding the result of interpreting inequalities helps in making sense of the solution's context. Thus, interpreting incorrectly solved inequalities helps students avoid misleading conclusions and sharpens their understanding of mathematical logic and analysis.