Certainly! Let's solve the inequality [tex]\(5 \leq 2r - 3 < 11\)[/tex] step by step.
### Step 1: Break the compound inequality into two parts.
We have two parts to consider in our compound inequality:
1. [tex]\(5 \leq 2r - 3\)[/tex]
2. [tex]\(2r - 3 < 11\)[/tex]
### Step 2: Solve the first inequality [tex]\(5 \leq 2r - 3\)[/tex].
#### Step 2.1: Isolate the term with [tex]\(r\)[/tex].
Add 3 to both sides:
[tex]\[
5 + 3 \leq 2r - 3 + 3
\][/tex]
This simplifies to:
[tex]\[
8 \leq 2r
\][/tex]
#### Step 2.2: Solve for [tex]\(r\)[/tex].
Divide both sides by 2:
[tex]\[
\frac{8}{2} \leq \frac{2r}{2}
\][/tex]
This simplifies to:
[tex]\[
4 \leq r
\][/tex]
So, one part of the solution is:
[tex]\[
r \geq 4
\][/tex]
### Step 3: Solve the second inequality [tex]\(2r - 3 < 11\)[/tex].
#### Step 3.1: Isolate the term with [tex]\(r\)[/tex].
Add 3 to both sides:
[tex]\[
2r - 3 + 3 < 11 + 3
\][/tex]
This simplifies to:
[tex]\[
2r < 14
\][/tex]
#### Step 3.2: Solve for [tex]\(r\)[/tex].
Divide both sides by 2:
[tex]\[
\frac{2r}{2} < \frac{14}{2}
\][/tex]
This simplifies to:
[tex]\[
r < 7
\][/tex]
So, the other part of the solution is:
[tex]\[
r < 7
\][/tex]
### Step 4: Combine the two parts of the solution.
From the above steps, we have:
1. [tex]\(r \geq 4\)[/tex]
2. [tex]\(r < 7\)[/tex]
Combining these, the solution to the compound inequality [tex]\(5 \leq 2r - 3 < 11\)[/tex] is:
[tex]\[
4 \leq r < 7
\][/tex]
Therefore, the values of [tex]\(r\)[/tex] that satisfy the inequality [tex]\(5 \leq 2r - 3 < 11\)[/tex] are in the interval:
[tex]\[
[4, 7)
\][/tex]
So, the complete solution to the inequality [tex]\(5 \leq 2r - 3 < 11\)[/tex] is:
[tex]\[
r \in [4, 7)
\][/tex]
