22. Solving Inequalities - Cardiff Tutor Company

Question 1

Solve the following inequalities:

1(a):

3x< 21

ANSWER: Multiple Choice (Type 1)

A: $x<7$

B: $x\leq 7$

C: $x<3$

D: $x\leq 3$

Answer: A

Workings:

$\dfrac{3x}{3}<\dfrac{21}{3}$

$x<7$

Marks = 1

1(b):

12+x>16

ANSWER: Multiple Choice (Type 1)

A: $x>6$

B: $x\geq 6$

C: $x>4$

D: $x\geq 4$

Answer: C

Workings:

$12+x-12>16-12$

$x>4$

Marks = 1

1(c):

7x\leq 14

ANSWER: Multiple Choice (Type 1)

A: $x\leq 7$

B: $x\leq 2$

C: $x\leq 14$

D: $x< 2$

Answer: B

Workings:

$\dfrac{7x}{7}\leq \dfrac{14}{7}$

$x\leq 2$

Marks = 1

1(d):

7x \geq 42

ANSWER: Multiple Choice (Type 1)

A: $x\geq 8$

B: $x> 8$

C: $x\geq 6$

D: $x> 6$

Answer: C

Workings:

$\dfrac{7x}{7}\geq \dfrac{42}{7}$

$x \geq 6$

Marks = 1

1(e):

7+x\leq 11

ANSWER = Multiple Choice (Type 1)

A: $x\leq -4$

B: $x\geq 4$

C: $-4\leq x$

D: $x\leq 4$

Answer: D

Workings:

$7+x-7\leq 11-7$

$x\leq 4$

Marks = 1

Question 2

Solve the following inequalities:

2(a):

2x+8<18

ANSWER: Simple Text Answer

A: $x<5$

B: $x>5$

C: $x<2$

D: $x>2$

Answer: A

Workings:

$2x+8-8<18-8$

$\dfrac{2x}{2}<\dfrac{10}{2}$

$x<5$

Marks = 1

2(b):

3x+10>13

ANSWER: Multiple Choice (Type 1)

A: $x>3$

B: $x<3$

C: $x>1$

D: $x<1$

Answer: C

Workings:

$3x+10-10>13-10$

$\dfrac{3x}{3}>\dfrac{3}{3}$

$x>1$

Marks = 1

2(c):

5x + 9 \leq 59

ANSWER: Multiple Choice (Type 1)

A: $x\leq 9$

B: $x< 9$

C: $x< 10$

D: $x\leq 10$

Answer: D

Workings:

$5x+9-9\leq 59-9$

$\dfrac{5x}{5}\leq \dfrac{50}{5}$

$x\leq 10$

Marks = 1

2(d):

10x+6\geq 56

ANSWER: Multiple Choice (Type 1)

A: $x\geq 6$

B: $x>6$

C: $x> 5$

D: $x\geq 5$

Answer: D

Workings:

$10x+6-6\geq 56-6$

$\dfrac{10x}{10} \geq \dfrac{50}{10}$

$x\geq 5$

Marks = 1

2(e):

9x + 14 \leq 95

ANSWER: Multiple Choice (Type 1)

A: $x< 9$

B: $x\leq 9$

C: $x\geq 9$

D: $x> 9$

Answer: B

Workings:

$9x+14-14\leq 95-14$

$\dfrac{9x}{9}\leq \dfrac{81}{9}$

$x\leq 9$

Marks = 1

Question 3

Solve the following inequalities:

3(a):

-7x + 3< -60

ANSWER: Multiple Choice (Type 1)

A: $x<9$

B: $x>9$

C: $x<3$

D: $x>3$

Answer: B

Workings:

$-7x+3-3<-60-3$

$\dfrac{-7x}{-7}<\dfrac{-63}{-7}$

$x>9$

Marks = 1

3(b):

3x+3>33

ANSWER: Simple Text Answer

A: $x>-10$

B: $x<-10$

C: $x<10$

D: $x>10$

Answer: D

Workings:

$3x+3-3>33-3$

$\dfrac{3x}{3}>\dfrac{30}{3}$

$x>10$

Marks = 1

3(c):

-2x - 7 \geq -7

ANSWER: Multiple Choice (Type 1)

A: $x \leq 0$

B: $x \geq 0$

C: $x \leq 7$

D: $x \geq 7$

Answer: A

Workings:

$-2x-7+7\geq -7+7$

$\dfrac{-2x}{-2}\geq \dfrac{0}{-2}$

$x \leq 0$

Marks = 1

3(d):

-10x + 13 \leq 13

ANSWER: Multiple Choice (Type 1)

A: $x\geq 0$

B: $x\leq 0$

C: $x> 0$

D: $x> 0$

Answer: A

Workings:

$-10x+13-13\leq 13-13$

$\dfrac{-10x}{-10} \leq \dfrac{0}{-10}$

$x\geq 0$

Marks = 1

Question 4

Solve the following inequalities

4(a):

\dfrac{4x-3}{2}>\dfrac{x+8}{4}

ANSWER: Multiple Choice (Type 1)

A: $x>2$

B: $x>-2$

C: $x>4$

D: $x>-4$

Answer: A

Workings:

$16x-12 > 2x+16$

$14x>28$

$x > 2$

Marks = 3

4(b):

\dfrac{2x-3}{2}>\dfrac{x+5}{4}

ANSWER: Multiple Choice (Type 1)

A: $x>\dfrac{13}{3}$

B: $x>{11}{2}$

C: $x>\dfrac{11}{3}$

D: $x>\dfrac{13}{2}$

Answer: C

Workings:

$8x-12>2x+10$

$6x>22$

$x>\dfrac{11}{3}$

Marks = 3

4(c):

\dfrac{3-x}{2}>\dfrac{x+3}{3}

ANSWER: Multiple Choice (Type 1)

A: $x>\dfrac{7}{3}$

B: $x< \dfrac{3}{5}$

C: $x< \dfrac{5}{6}$

D: $x> \dfrac{3}{2}$

Answer: B

Workings:

$9-3x>2x+6$

$-5x>-3$

$x<\dfrac{3}{5}$

Marks = 3

Question 5

Solve the following inequalities

5(a)

-1<\dfrac{5x-5}{3}<5

ANSWER: Multiple Choice (Type 1)

A: $\dfrac{7}{3}<x<3$

B: $\dfrac{5}{4}<x<5$

C: $\dfrac{4}{3}<x<3$

D: $\dfrac{2}{5}<x<4$

Answer: D

Workings:

$-3<5x-5<15$

$2<5x<20$

$\dfrac{2}{5} < x < 4$

Marks = 3

5(b):

5<\dfrac{3x-5}{2}<9

ANSWER: Multiple Choice (Type 1)

A: $7<x<\dfrac{26}{3}$

B: $6<x<9$

C: $\dfrac{29}{4}<x<\dfrac{37}{4}$

D: $5<x<\dfrac{23}{3}$

Answer: D

Workings:

$10<3x-5<18$

$15<3x<23$

$5<x<\dfrac{23}{3}$

Marks = 3

5(c):

-4 <7-3x <5

ANSWER: Multiple Choice (Type 1)

A: $\dfrac{3}{5}<x<\dfrac{7}{4}$

B: $\dfrac{9}{7}>x>\dfrac{3}{4}$

C: $\dfrac{11}{3}>x>\dfrac{2}{3}$

D: $\dfrac{5}{7}<x<\dfrac{5}{2}$

Answer: C

Workings:

$-11<-3x<-2$

$\dfrac{11}{3}>x>\dfrac{2}{3}$

Marks = 3