Question

Find the range of \(x\) satisfying both: \(|2x - 7|<5\) and \(x + 3>0\)

• A. \(1<x<6\)
• B. \(x>1\)
• C. \(1<x<7\)
• D. \(x>-3\)

Hint:

When finding the intersection of two or more inequalities, it's often helpful to visualize the solution sets on a number line. This makes it easy to see the overlapping region that represents the final answer.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem requires us to find the set of all values of \(x\) that simultaneously satisfy two given inequalities. This means we must solve each inequality separately and then find the intersection of their solution sets.
Step 2: Key Formula or Approach:
- For an absolute value inequality of the form \(|A|<B\) (where \(B>0\)), the solution is given by \(-B<A<B\).
- For a simple linear inequality, we isolate the variable \(x\) using standard algebraic operations.
Step 3: Detailed Explanation:
First, we solve the absolute value inequality:
\[ |2x - 7|<5 \] Using the rule for absolute value inequalities, we can rewrite this as:
\[ -5<2x - 7<5 \] To solve for \(x\), we first add 7 to all three parts of the inequality:
\[ -5 + 7<2x - 7 + 7<5 + 7 \] \[ 2<2x<12 \] Next, we divide all three parts by 2:
\[ \frac{2}{2}<\frac{2x}{2}<\frac{12}{2} \] \[ 1<x<6 \] So, the solution to the first inequality is the open interval (1, 6).
Second, we solve the linear inequality:
\[ x + 3>0 \] Subtract 3 from both sides:
\[ x>-3 \] The solution to the second inequality is the interval (\(-3, \infty\)).
Finally, we need to find the range of \(x\) that satisfies both conditions. We are looking for the intersection of the two solution sets: \(x \in (1, 6)\) AND \(x \in (-3, \infty)\).
The numbers that are both greater than 1 AND less than 6, and also greater than -3, are simply the numbers between 1 and 6.
Therefore, the intersection is \(1<x<6\).
Step 4: Final Answer
The range of \(x\) satisfying both inequalities is \(1<x<6\).