Question

to solve inequalities involving addition, use the subtraction property of inequality. solve: ( p + 7 leq 9 ) ( p + 7 - 7 leq 9 - 7 ) ← subtract 7 from both sides. ( p leq 2 ) ← simplify the solution set contains numbers less than or equal to 2. check: according to the solution set, 2 is a solution and 4 is not. let ( p = 2 ). ( 2 + 7 stackrel{?}{leq} 9 ) ( 9 leq 9 ) true let ( p = 4 ). ( 4 + 7 stackrel{?}{leq} 9 ) ( 11 leq 9 ) false to solve inequalities involving subtraction, use the addition property of inequality. graph the solution set for ( p leq 2 ). remember: the graph of an inequality containing ( < ) or ( leq ) has a ray that points left. the graph of an inequality containing ( > ) or ( geq ) has a ray that points right. (overleftarrow{quadquadquadquad\bulletquadquad}) (-4) (-3) (-2) (-1) (0) (1) (2) (3) check: according to the solution set, both 2 and 0 are solutions. so there must be a dot on point 2 and the ray must point left. so the graph is correct. solve the inequality. check to justify your answer. then write the solution set, ( s ), using words. 1. ( n + 4 leq 10 ) ( n + 4 - 4 leq 10 - 4 ) ( n leq 6 ) check: ( 6 + 4 leq 10 ); ( 10 leq 10 ) ( s = {\text{numbers less than or equal to 6}} ) 2. ( -9 + b leq 45 ) 3. ( x - 4 geq -11 ) 4. ( 3 + h geq 31 ) 5. ( -13 > k - 8 ) 6. ( 24 > f - 6 ) combine terms to solve. check to justify your answer. then write the solution set, ( s ), using words. 7. ( (8 - 2) + t > 8 ) ( (6) + t > 8 ); ( 6 - 6 + t > 8 - 6 ) ( t > 2 ) ( s = {\text{numbers greater than 2}} ) 8. ( (6 - 5) + g > 1 ) 9. ( 5 geq j + (2 - 4) ) 10. ( -4 geq p + (-2 - 1) ) 11. ( -22 < t - 3 - 1 ) 12. ( -1 < b - 10 + (-2) )

Explanation:

Step1: Solve inequality 2

Add 9 to both sides.
$$-9 + b + 9 \leq 45 + 9$$
$$b \leq 54$$

Step2: Check inequality 2

Test $b=54$: $-9+54=45$, $45\leq45$ (True). Test $b=55$: $-9+55=46$, $46\leq45$ (False).

Step3: Solve inequality 3

Add 4 to both sides.
$$x - 4 + 4 \geq -11 + 4$$
$$x \geq -7$$

Step4: Check inequality 3

Test $x=-7$: $-7-4=-11$, $-11\geq-11$ (True). Test $x=-8$: $-8-4=-12$, $-12\geq-11$ (False).

Step5: Solve inequality 4

Subtract 3 from both sides.
$$3 + h - 3 \geq 31 - 3$$
$$h \geq 28$$

Step6: Check inequality 4

Test $h=28$: $3+28=31$, $31\geq31$ (True). Test $h=27$: $3+27=30$, $30\geq31$ (False).

Step7: Solve inequality 5

Add 8 to both sides.
$$-13 + 8 \geq k - 8 + 8$$
$$-5 \geq k \text{ or } k \leq -5$$

Step8: Check inequality 5

Test $k=-5$: $-5-8=-13$, $-13\geq-13$ (True). Test $k=-4$: $-4-8=-12$, $-13\geq-12$ (False).

Step9: Solve inequality 6

Add 6 to both sides.
$$24 + 6 > f - 6 + 6$$
$$30 > f \text{ or } f < 30$$

Step10: Check inequality 6

Test $f=30$: $30-6=24$, $24>24$ (False). Test $f=29$: $29-6=23$, $24>23$ (True).

Step11: Solve inequality 8

Simplify, subtract 1 from both sides.
$$1 + g > 1$$
$$1 + g - 1 > 1 - 1$$
$$g > 0$$

Step12: Check inequality 8

Test $g=0$: $1+0=1$, $1>1$ (False). Test $g=1$: $1+1=2$, $2>1$ (True).

Step13: Solve inequality 9

Simplify, add 2 to both sides.
$$5 \geq j - 2$$
$$5 + 2 \geq j - 2 + 2$$
$$7 \geq j \text{ or } j \leq 7$$

Step14: Check inequality 9

Test $j=7$: $7+(2-4)=5$, $5\geq5$ (True). Test $j=8$: $8+(2-4)=6$, $5\geq6$ (False).

Step15: Solve inequality 10

Simplify, add 3 to both sides.
$$-4 \geq p - 3$$
$$-4 + 3 \geq p - 3 + 3$$
$$-1 \geq p \text{ or } p \leq -1$$

Step16: Check inequality 10

Test $p=-1$: $-1+(-2-1)=-4$, $-4\geq-4$ (True). Test $p=0$: $0+(-2-1)=-3$, $-4\geq-3$ (False).

Step17: Solve inequality 11

Simplify, add 4 to both sides.
$$-22 < t - 4$$
$$-22 + 4 < t - 4 + 4$$
$$-18 < t \text{ or } t > -18$$

Step18: Check inequality 11

Test $t=-18$: $-18-3-1=-22$, $-22<-22$ (False). Test $t=-17$: $-17-3-1=-21$, $-22<-21$ (True).

Step19: Solve inequality 12

Simplify, add 8 to both sides.
$$-1 < b - 8$$
$$-1 + 8 < b - 8 + 8$$
$$7 < b \text{ or } b > 7$$

Step20: Check inequality 12

Test $b=7$: $7-10+(-2)=-5$, $-1<-5$ (False). Test $b=8$: $8-10+(-2)=-4$, $-1<-4$ (False, correction: $8-10+(-2)=-4$, $-1>-4$, so test $b=8$: $-1 < -4$ is False, test $b=9$: $9-10+(-2)=-3$, $-1 < -3$ (False, correction: $9-10+(-2)=-3$, $-1 > -3$, so correct test: $b=8$: $8-10+(-2)=-4$, $-1 < -4$ (False); $b=6$: $6-10+(-2)=-6$, $-1 < -6$ (False). Correct check: $b=8$: $-1 < 8 - 10 + (-2) \to -1 < -4$ (False, so $b>7$ means test $b=8$ does not satisfy, test $b=9$: $-1 < 9-10+(-2) \to -1 < -3$ (False, error in simplification: $10+(-2)=8$, so $b - 8$, so $-1 < b - 8 \to b > 7$. Test $b=8$: $8-8=0$, $-1 < 0$ (True). My earlier simplification was wrong: $b - [10+(-2)] = b - 8$. Correct check: $b=8$: $-1 < 0$ (True); $b=7$: $-1 < -1$ (False).

Answer:

1. $n \leq 6$; $S=$ {numbers less than or equal to 6}
2. $b \leq 54$; $S=$ {numbers less than or equal to 54}
3. $x \geq -7$; $S=$ {numbers greater than or equal to -7}
4. $h \geq 28$; $S=$ {numbers greater than or equal to 28}
5. $k \leq -5$; $S=$ {numbers less than or equal to -5}
6. $f < 30$; $S=$ {numbers less than 30}
7. $t > 2$; $S=$ {numbers greater than 2}
8. $g > 0$; $S=$ {numbers greater than 0}
9. $j \leq 7$; $S=$ {numbers less than or equal to 7}
10. $p \leq -1$; $S=$ {numbers less than or equal to -1}
11. $t > -18$; $S=$ {numbers greater than -18}
12. $b > 7$; $S=$ {numbers greater than 7}