To determine the solution set of the compound inequality [tex]\(1.5x - 1 > 6.5\)[/tex] or [tex]\(7x + 3 < -25\)[/tex], we need to solve each inequality individually and then combine the solutions.
### Step 1: Solve the first inequality [tex]\(1.5x - 1 > 6.5\)[/tex]
1. Isolate the term with the variable:
Add 1 to both sides of the inequality:
[tex]\[
1.5x - 1 + 1 > 6.5 + 1
\][/tex]
Simplifies to:
[tex]\[
1.5x > 7.5
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
Divide both sides by 1.5:
[tex]\[
x > \frac{7.5}{1.5}
\][/tex]
Simplifies to:
[tex]\[
x > 5
\][/tex]
### Step 2: Solve the second inequality [tex]\(7x + 3 < -25\)[/tex]
1. Isolate the term with the variable:
Subtract 3 from both sides of the inequality:
[tex]\[
7x + 3 - 3 < -25 - 3
\][/tex]
Simplifies to:
[tex]\[
7x < -28
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
Divide both sides by 7:
[tex]\[
x < \frac{-28}{7}
\][/tex]
Simplifies to:
[tex]\[
x < -4
\][/tex]
### Step 3: Combine the Solutions
The compound inequality [tex]\(1.5x - 1 > 6.5\)[/tex] or [tex]\(7x + 3 < -25\)[/tex] is satisfied if either of the individual inequalities is true.
- The solution to [tex]\(1.5x - 1 > 6.5\)[/tex] is:
[tex]\[
x > 5
\][/tex]
- The solution to [tex]\(7x + 3 < -25\)[/tex] is:
[tex]\[
x < -4
\][/tex]
### Graphing the Solution Set
- [tex]\(x > 5\)[/tex]: This represents all values of [tex]\(x\)[/tex] greater than 5.
- [tex]\(x < -4\)[/tex]: This represents all values of [tex]\(x\)[/tex] less than -4.
### Conclusion
The graph of the solution set will have two disjoint parts:
1. A part where [tex]\(x > 5\)[/tex].
2. Another part where [tex]\(x < -4\)[/tex].
In interval notation, the solution set is:
[tex]\[
(-\infty, -4) \cup (5, \infty)
\][/tex]
So, the correct graph will show two separate regions on the number line: one to the left of [tex]\(-4\)[/tex] (but not including [tex]\(-4\)[/tex]), and one to the right of 5 (but not including 5).
