Question

find the solution set for each inequality.
inequality replacement set, r solution set, s

1. $x > 8$ {2, 4, 6, 8, 10, 12} {10, 12}
2. $a < 16$ {14, 15, 16, 17}
3. $jgeq9$ {1, 10, 100}
4. $cleq90$ {40, 50, 60, 70, 80, ...}
5. $dgeq - 1$ {-3, -2, -1, 0, 1, 2, 3}
6. $g < 12$ {-4, -2, 0, 2, 4, 6, ...}
7. $h > 10$ {10, 20, 30, 40, ...}
8. $k < 6$ {positive multiples of 4}
9. $l > 100$ {whole numbers}
10. $f < 60$ {60, 61, 62, 63, ...}
11. $n + 2>20$ {2, 4, 6, 8, 10}
12. $p - 4geq10$ {9, 13, 17, 21}
13. $5t > - 15$ {-4, -3, -2, -1, 0, 1, 2}
14. $\frac{g}{-3}leq4$ {-24, -18, -12, -6, 0, 6}

Explanation:

Step1: Check each element in replacement set for inequality 2

For the inequality $a < 16$ and replacement set $\{14,15,16,17\}$, we test each number.
$14<16$, $15 < 16$, $16
ot<16$, $17
ot<16$.

Step2: Determine solution set

The solution set consists of the numbers that satisfy the inequality, so $S=\{14,15\}$.

Step3: Check each element in replacement set for inequality 3

For the inequality $j\geq9$ and replacement set $\{1,10,100\}$, we test each number.
$1
ot\geq9$, $10\geq9$, $100\geq9$. So $S = \{10,100\}$.

Step4: Check each element in replacement set for inequality 4

For the inequality $c\leq90$ and replacement set $\{40,50,60,70,80,\cdots\}$, all numbers in the set satisfy the inequality. So $S=\{40,50,60,70,80,\cdots\}$.

Step5: Check each element in replacement set for inequality 5

For the inequality $d\geq - 1$ and replacement set $\{-3,-2,-1,0,1,2,3\}$, we test each number.
$-3
ot\geq - 1$, $-2
ot\geq - 1$, $-1\geq - 1$, $0\geq - 1$, $1\geq - 1$, $2\geq - 1$, $3\geq - 1$. So $S=\{-1,0,1,2,3\}$.

Step6: Check each element in replacement set for inequality 6

For the inequality $g < 12$ and replacement set $\{-4,-2,0,2,4,6,\cdots\}$, all numbers in the set satisfy the inequality. So $S=\{-4,-2,0,2,4,6,\cdots\}$.

Step7: Check each element in replacement set for inequality 7

For the inequality $h>10$ and replacement set $\{10,20,30,40,\cdots\}$, $10
ot>10$, $20>10$, $30>10$, $40>10,\cdots$. So $S=\{20,30,40,\cdots\}$.

Step8: Check each element in replacement set for inequality 8

For the inequality $k < 6$ and replacement set $\{\text{positive multiples of }4\}=\{4,8,12,\cdots\}$, $4<6$, $8
ot<6$, $12
ot<6,\cdots$. So $S = \{4\}$.

Step9: Check each element in replacement set for inequality 9

For the inequality $l>100$ and replacement set $\{\text{whole numbers}\}$, the solution set is $\{101,102,103,\cdots\}$.

Step10: Check each element in replacement set for inequality 10

For the inequality $f < 60$ and replacement set $\{60,61,62,63,\cdots\}$, no number in the set satisfies the inequality. So $S=\varnothing$.

Step11: Check each element in replacement set for inequality 11

For the inequality $n + 2>20$ or $n>18$ and replacement set $\{2,4,6,8,10\}$, no number in the set satisfies the inequality. So $S=\varnothing$.

Step12: Check each element in replacement set for inequality 12

For the inequality $p-4\geq10$ or $p\geq14$ and replacement set $\{9,13,17,21\}$, $9
ot\geq14$, $13
ot\geq14$, $17\geq14$, $21\geq14$. So $S=\{17,21\}$.

Step13: Check each element in replacement set for inequality 13

For the inequality $5t>-15$ or $t > - 3$ and replacement set $\{-4,-3,-2,-1,0,1,2\}$, $-4
ot>-3$, $-3
ot>-3$, $-2>-3$, $-1>-3$, $0>-3$, $1>-3$, $2>-3$. So $S=\{-2,-1,0,1,2\}$.

Step14: Check each element in replacement set for inequality 14

For the inequality $\frac{g}{-3}\leq4$ or $g\geq - 12$ (after multiplying both sides by - 3 and reversing the inequality sign) and replacement set $\{-24,-18,-12,-6,0,6\}$, $-24
ot\geq - 12$, $-18
ot\geq - 12$, $-12\geq - 12$, $-6\geq - 12$, $0\geq - 12$, $6\geq - 12$. So $S=\{-12,-6,0,6\}$.

Answer:

1. $\{14,15\}$
2. $\{10,100\}$
3. $\{40,50,60,70,80,\cdots\}$
4. $\{-1,0,1,2,3\}$
5. $\{-4,-2,0,2,4,6,\cdots\}$
6. $\{20,30,40,\cdots\}$
7. $\{4\}$
8. $\{101,102,103,\cdots\}$
9. $\varnothing$
10. $\varnothing$
11. $\{17,21\}$
12. $\{-2,-1,0,1,2\}$
13. $\{-12,-6,0,6\}$