To graph the solution to the inequalities [tex]\(3x - 5 < -14\)[/tex] or [tex]\(2x - 1 > 7\)[/tex] on the number line, follow these steps:
1. Solve the inequalities separately:
- For the inequality [tex]\(3x - 5 < -14\)[/tex]:
[tex]\[
3x - 5 < -14
\][/tex]
Adding 5 to both sides:
[tex]\[
3x < -9
\][/tex]
Dividing both sides by 3:
[tex]\[
x < -3
\][/tex]
- For the inequality [tex]\(2x - 1 > 7\)[/tex]:
[tex]\[
2x - 1 > 7
\][/tex]
Adding 1 to both sides:
[tex]\[
2x > 8
\][/tex]
Dividing both sides by 2:
[tex]\[
x > 4
\][/tex]
2. Combine the solutions:
The solution to [tex]\(3x - 5 < -14\)[/tex] is [tex]\(x < -3\)[/tex].
The solution to [tex]\(2x - 1 > 7\)[/tex] is [tex]\(x > 4\)[/tex].
Since we are dealing with an "or" condition, we take the union of both solutions. This means we combine all [tex]\(x\)[/tex] that satisfy either [tex]\(x < -3\)[/tex] or [tex]\(x > 4\)[/tex].
3. Graph the solution on a number line:
On a number line:
- Shade the region to the left of [tex]\(x = -3\)[/tex] with an open circle at [tex]\(-3\)[/tex], indicating that [tex]\(-3\)[/tex] is not included in the solution.
- Shade the region to the right of [tex]\(x = 4\)[/tex] with an open circle at [tex]\(4\)[/tex], indicating that [tex]\(4\)[/tex] is not included in the solution.
Here is the visual representation on the number line:
[tex]\[
\begin{array}{ccccccccccccccccccc}
\text{<--} & \text{•} & & & & & & & & & & & & & & \text{•} & \text{-->} \\
& -3 & & & & 0 & & & & & & & & 4 & &
\end{array}
\][/tex]
Note:
- The circles at [tex]\(-3\)[/tex] and [tex]\(4\)[/tex] indicate that these points are not included in the solution (denoted as open circles).
- The shaded lines extending to the left of [tex]\(-3\)[/tex] and to the right of [tex]\(4\)[/tex] indicate the solution intervals.
