Let's solve each inequality step-by-step:
1. Solving the first inequality: [tex]\( 2x - 17 < -14 \)[/tex]
- Step 1: Add 17 to both sides of the inequality to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
2x - 17 + 17 < -14 + 17
\][/tex]
[tex]\[
2x < 3
\][/tex]
- Step 2: Divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
x < \frac{3}{2}
\][/tex]
[tex]\[
x < 1.5
\][/tex]
2. Solving the second inequality: [tex]\( 2x - 17 > 1.4 \)[/tex]
- Step 1: Add 17 to both sides of the inequality to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
2x - 17 + 17 > 1.4 + 17
\][/tex]
[tex]\[
2x > 18.4
\][/tex]
- Step 2: Divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
x > \frac{18.4}{2}
\][/tex]
[tex]\[
x > 9.2
\][/tex]
Now, we combine the solutions from both inequalities. The solution set consists of the union of the two intervals:
- From the first inequality: [tex]\( x < 1.5 \)[/tex]
- From the second inequality: [tex]\( x > 9.2 \)[/tex]
The combined solution set in interval notation is:
[tex]\[
(-\infty, 1.5) \cup (9.2, \infty)
\][/tex]
So the solution set is:
[tex]\[
(-\infty, 1.5) \cup (9.2, \infty)
\][/tex]
Next, let's graph the solution set on a number line:
- For [tex]\( x < 1.5 \)[/tex], we use an open circle at [tex]\( 1.5 \)[/tex] and shade to the left.
- For [tex]\( x > 9.2 \)[/tex], we use an open circle at [tex]\( 9.2 \)[/tex] and shade to the right.
Among the choices provided, you should choose the graph that best represents this solution.
Given the description, the correct graph (which might be labeled as choice A or C) would have open circles at [tex]\( 1.5 \)[/tex] and [tex]\( 9.2 \)[/tex] with shading to the left of [tex]\( 1.5 \)[/tex] and to the right of [tex]\( 9.2 \)[/tex].
Therefore, the correct interval notation is:
[tex]\[
(-\infty, 1.5) \cup (9.2, \infty)
\][/tex]
The graphical solution should show the appropriate intervals as described.
