Answer :
To solve the compound inequality [tex]\(14 > 17f - 3 \geq -20\)[/tex], we will break it into two separate inequalities and solve for [tex]\(f\)[/tex] in each of them.
1. Solve the left part of the inequality:
[tex]\(14 > 17f - 3\)[/tex]
- First, add 3 to both sides to isolate the term with [tex]\(f\)[/tex]:
[tex]\(14 + 3 > 17f\)[/tex]
[tex]\(17 > 17f\)[/tex]
- Next, divide both sides by 17 to solve for [tex]\(f\)[/tex]:
[tex]\(f < \frac{17}{17}\)[/tex]
[tex]\(f < 1\)[/tex]
2. Solve the right part of the inequality:
[tex]\(17f - 3 \geq -20\)[/tex]
- First, add 3 to both sides to isolate the term with [tex]\(f\)[/tex]:
[tex]\(17f \geq -20 + 3\)[/tex]
[tex]\(17f \geq -17\)[/tex]
- Next, divide both sides by 17 to solve for [tex]\(f\)[/tex]:
[tex]\(f \geq \frac{-17}{17}\)[/tex]
[tex]\(f \geq -1\)[/tex]
3. Combine the results:
The solution for the compound inequality is [tex]\(-1 \leq f < 1\)[/tex].
So, [tex]\(f\)[/tex] is in the range [tex]\([-1, 1)\)[/tex].
1. Solve the left part of the inequality:
[tex]\(14 > 17f - 3\)[/tex]
- First, add 3 to both sides to isolate the term with [tex]\(f\)[/tex]:
[tex]\(14 + 3 > 17f\)[/tex]
[tex]\(17 > 17f\)[/tex]
- Next, divide both sides by 17 to solve for [tex]\(f\)[/tex]:
[tex]\(f < \frac{17}{17}\)[/tex]
[tex]\(f < 1\)[/tex]
2. Solve the right part of the inequality:
[tex]\(17f - 3 \geq -20\)[/tex]
- First, add 3 to both sides to isolate the term with [tex]\(f\)[/tex]:
[tex]\(17f \geq -20 + 3\)[/tex]
[tex]\(17f \geq -17\)[/tex]
- Next, divide both sides by 17 to solve for [tex]\(f\)[/tex]:
[tex]\(f \geq \frac{-17}{17}\)[/tex]
[tex]\(f \geq -1\)[/tex]
3. Combine the results:
The solution for the compound inequality is [tex]\(-1 \leq f < 1\)[/tex].
So, [tex]\(f\)[/tex] is in the range [tex]\([-1, 1)\)[/tex].
