23. Quadratic Inequalities (Algebraically)

Question 1

Consider the following inequalities.

1(a):

For which values of $x$ is the following inequality true?

x^2-3x+4>2

ANSWER: Multiple Choice (Type 1)

A: $x=0$ & $x=1$

B: $x=1$ & $x=3$

C: $x=1$ & $x=2$

D: $x=0$ & $x=3$

Answer: C

Workings:

When substituting $x$ values into the inequality, only $x=1$ and $x=2$ give a valid inequality.

Marks = 1

1(b):

For which inequality is the value of $x$ true?

x=7

ANSWER: Multiple Choice (Type 1)

A: $x^2-7x+7<7$

B: $-x^2+7x+7<7$

C: $x^2+7x-7<7$

D: $-x^2+7x-7<7$

Answer: D

Workings:

When substituting $x=7$ into the inequalities, only D gives a valid inequality.

Marks = 1

1(c):

Which solutions satisfy the following inequality?

x^2+7x-30<0

ANSWER: Multiple Choice (Type 1)

A: $3<x<4$

B: $-10<x<3$

C: $-3<x<10$

D: $-5<x<6$

Answer: B

Workings:

Putting $x^2+7x-30=0$

$(x+10)(x-3)=0$

This is true when $x=-10$ and $x=3$ .

Because $x^2$ is positive, the value of the equation will be negative between the points $x=-10$ and $x=3$ .

So the inequality is satisfied by the solutions $-10<x<3$ .

Marks = 1

Question 2

Solve the following inequalities:

2(a):

x^2+5x-13\leq1

ANSWER: Multiple Choice (Type 1)

A: $-4\leq x\leq 1$

B: $0\leq x\leq 5$

C: $-7\leq x\leq 2$

D: $-3\leq x\leq 3$

Answer: C

Workings:

$x^2+5x-14\leq 0$

$(x+7)(x-2)\leq 0$

This occurs when $x$ is between -7 & 2

$-7\leq x\leq 2$

Marks = 2

2(b):

x^2-10x+16 \leq 0

ANSWER: Multiple Choice (Type 1)

A: $4\leq x\leq 8$

B: $2\leq x\leq 8$

C: $1< x\leq 5$

D: $3\leq x< 5$

Answer: B

Workings:

$(x-8)(x-2)\leq 0$

This occurs when $x$ is between 2 & 8.

$2\leq x\leq 8$

Marks = 2

2(c):

x² > 4(8 - x)

ANSWER: Multiple Choice (Type 1)

A: $-8<x<4$

B: $-4\leq x\leq 4$

C: $-2<x\leq 2$

D: $-2\leq x<4$

Answer: A

Workings:

$x^2+4x-32>0$

$(x+8)(x-4)>0$

This occurs when $x$ is between -8 & 4.

$-8<x<4$

Marks = 2

2(d):

x² - x - 30 \leq 0

ANSWER: Multiple Choice (Type 1)

A: $-1\leq x\leq 7$

B: $-9\leq x\leq 4$

C: $-4\leq x\leq 3$

D: $-5\leq x\leq 6$

Answer: D

Workings:

(x-6)(x+5)\leq 0

This occurs when $x$ is between -5 & 6.

-5\leq x \leq 6

Marks = 2

Question 3

Donald and Amir disagree about the solution to the inequality,

x^2-4x-13\geq-8

Donald claims that the solution is $x\geq 5$

Amir states that the solution is $x\leq -1$

Who is correct?

ANSWER: Multiple Choice (Type 1)

A: Amir

B: Donald

C: Both

D: Neither

Answer: C

Workings:

$x^2-4x-5\geq 0$

$(x-5)(x+1)\geq 0$

This occurs when $x$ is more than 5 or less than -1.

So Amir and Donald are BOTH correct as they have each stated half of the full inequality.

Marks = 3

Question 4

For the following inequality,

-x^2+7x-12\geq0

determine if the solution is,

$3\leq x \leq4$ or $3\geq x\geq4$ .

ANSWER: Multiple Choice (Type 1)

A: $1\leq x\leq 5$

B: $-1\leq x\leq 1$

C: $3\leq x\leq 4$

D: $0\leq x\leq 3$

Answer: C

Workings:

$-(x^2-7x+12)\geq 0$

$-(x-4)(x-3)\geq 0$

Because we have a negative quadratic. this must occur between $x=3$ & $x=4$ .

$3\leq x\leq 4$

Marks = 4

Question 5

Solve for the following inequality,

x^2-9x-5\leq-4x-9

ANSWER: Multiple Choice (Type 1)

A: $-1\leq x\leq 4$

B: $0\leq x\leq 4$

C: $1\leq x\leq 4$

D: $2\leq x\leq 4$

Answer: C

Workings:

$x^2-5x+4\leq 0$

$(x-4)(x-1)\leq 0$

This occurs when $x$ is between 1 & 4.

$1\leq x\leq 4$

Marks = 3