Question
1.1 Given the following linear inequality: \( -6<2 x-4 \leq 2 \) 1.1.1 Solve the linear inequality.
Ask by Lambert Elliott. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution to the inequality is \( -1 < x \leq 3 \).
Solution
To solve the linear inequality:
\[
-6 < 2x - 4 \leq 2
\]
**Step 1: Isolate the term with \( x \)**
Add 4 to all parts of the inequality to eliminate the constant term on the right side of the \( 2x \):
\[
-6 + 4 < 2x - 4 + 4 \leq 2 + 4
\]
Simplifying each part:
\[
-2 < 2x \leq 6
\]
**Step 2: Solve for \( x \)**
Divide all parts of the inequality by 2 to isolate \( x \):
\[
\frac{-2}{2} < \frac{2x}{2} \leq \frac{6}{2}
\]
Simplifying:
\[
-1 < x \leq 3
\]
**Solution:**
The set of all real numbers \( x \) that satisfy the inequality is:
\[
-1 < x \leq 3
\]
In interval notation, this is written as:
\[
(-1,\ 3]
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To solve the linear inequality \( -6<2x-4 \leq 2 \), we can break it down into two parts. First, we solve the left side of the inequality, \( -6 < 2x - 4 \): 1. Add 4 to both sides: \( -6 + 4 < 2x \) which simplifies to \( -2 < 2x \). 2. Divide each side by 2: \( -1 < x \), or \( x > -1 \). Next, we solve the right side of the inequality, \( 2x - 4 \leq 2 \): 1. Add 4 to both sides: \( 2x \leq 6 \). 2. Divide each side by 2: \( x \leq 3 \). Now we combine both parts: \( -1 < x \leq 3 \). Therefore, the solution to the inequality is \( -1 < x \leq 3 \), which can also be expressed in interval notation as \( (-1, 3] \).
