Question

name
date
write each word phrase as an algebraic inequality.

1. 3 more than twice a number, n, is at most 9
2. ten less than a number, n, is at least 14

find the solution sets, given the replacement set ( r = {1, 2, 3, 4, 5} ).

1. ( p>3 )
2. ( c<6 )
3. ( a + 1<5 )
4. ( 4geq d - 1 )

graph each inequality on a number line.

1. ( e>3 )
2. ( dleq - 7 )
3. ( 4leq b )

write an inequality for each graph.
10.
11.
12.
tell whether each statement is true or false for the given value of the variable.

1. ( d - 7leq 6 ) when ( d=-11 )
2. ( 3(4 + x)>30 ) when ( x = 6 )
3. ( c + 6>13 ) when ( c = 4 )

solve each inequality. check to justify your work.

1. ( 6b<126 )
2. ( -\frac{d}{3}leq 9 )
3. ( y - 18>-10 )
4. ( 33 - xgeq 14 )
5. ( \frac{c}{4}+3>1 )
6. ( -3 - 4tgeq 5 )
7. ( 4g - 26>10 )
8. ( 2-\frac{a}{7}>-6 )

Explanation:

1. Translate phrase to inequality

"3 more than twice n is at most 9" → $2n + 3 \leq 9$

2. Translate phrase to inequality

"10 less than n is at least 14" → $n - 10 \geq 14$

3. Find p > 3 in {1,2,3,4,5}

Numbers >3: 4,5 → {4,5}

4. Find c <6 in {1,2,3,4,5}

All numbers <6 → {1,2,3,4,5}

5. Simplify a+1 <5 → a<4; find in set

Numbers <4:1,2,3 → {1,2,3}

6. Simplify 4≥d-1 → d≤5; find in set

All numbers ≤5 → {1,2,3,4,5}

7. Graph e>3

Open circle at 3, shade right

8. Graph d≤-7

Closed circle at -7, shade left

9. Rewrite 4≤b → b≥4; graph

Closed circle at 4, shade right

10. Graph has closed dot at 1/5, shade left

$x \leq \frac{1}{5}$

11. Graph has open dot at -3, shade left

$x < -3$

12. Graph has open dot at 0.5, shade right

$x > 0.5$
##13. Substitute d=-11: -11-7=-18≤6 → True
##14. Substitute x=6:3(4+6)=30, not >30 → False
##15. Substitute c=4:4+6=10>13? No → False
##16. Divide by 6: b<21
##17. Multiply by -3 (reverse inequality): d≥-27
##18. Add 18: y>8
##19. Subtract 33, multiply by -1: x≤19
##20. Subtract 3, multiply by4: c>-8
##21. Add3, divide by-4 (reverse): t≤-2
##22. Add26, divide by4: g>9
##23. Subtract2, multiply by-7 (reverse): a<56

Answer:

1. $2n + 3 \leq 9$
2. $n - 10 \geq 14$
3. $\{4,5\}$
4. $\{1,2,3,4,5\}$
5. $\{1,2,3\}$
6. $\{1,2,3,4,5\}$
7. Open circle at 3, shade right
8. Closed circle at -7, shade left
9. Closed circle at 4, shade right
10. $x \leq \frac{1}{5}$
11. $x < -3$
12. $x > 0.5$
13. True
14. False
15. False
16. $b < 21$
17. $d \geq -27$
18. $y > 8$
19. $x \leq 19$
20. $c > -8$
21. $t \leq -2$
22. $g > 9$
23. $a < 56$