Question
1.2 Given the following linear inequality: \( -x \leq 2 \) 1.2.1 Solve the linear inequality.
Ask by Cross Schofield. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x \geq -2 \)
Solution
To solve the linear inequality:
\[
-x \leq 2
\]
**Step 1: Isolate the variable \( x \).**
To isolate \( x \), we need to eliminate the negative sign in front of it. We can do this by multiplying both sides of the inequality by \(-1\). However, it's important to remember that multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.
\[
-x \leq 2 \quad \text{(Original Inequality)}
\]
\[
(-1)(-x) \geq (-1)(2) \quad \text{(Multiply both sides by -1 and reverse the inequality)}
\]
\[
x \geq -2
\]
**Solution:**
\[
x \geq -2
\]
**Interpretation:**
All real numbers \( x \) that are greater than or equal to \(-2\) satisfy the inequality. In interval notation, the solution can be expressed as:
\[
x \in [-2, \infty)
\]
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Simplify this solution Beyond the Answer
To solve the linear inequality \( -x \leq 2 \), first, we can multiply both sides by -1. Remember that when you multiply or divide by a negative number, you must flip the inequality sign. Thus, the inequality becomes \( x \geq -2 \). So, the solution set is all values of \( x \) that are greater than or equal to -2, which can be expressed in interval notation as \( [-2, \infty) \). Now you can visualize it on a number line, with -2 included and everything to the right extending infinitely!
