11 ALGEBRA

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Университет имени Сулеймана Демиреля

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A) x < 0 or x > 2

B) –2 < x < 1

C) x < –2 or x > 1

D) 0 < x < 2

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			x	–¥	1		1		2	¥
					1
					2

			(1)	+		+		–		+

		Case 1	(2)	+		–		–		+

			system

			(1)	+		+		–		+

		Case 2	(2)	–		–		–		+

			system
1
1			{2}, i.e.
So x	2	, 1 (1, 2)	{2}, i.e.
	2
1
x	2	, 2 .
	2

Check Yourself 4

Solve the systems.

	x2	– 4x 0		x2 4x+ 3 > 0			3. x2 x >1+ x
1.			2.				3. x2 x >1+ x
	x – 3		0	2	2
				x	(x 7) > 0

Answers
1.	[4, )	2.	x 0, x 7, (– ,	1) (3, ) 3.		( ,	)
							3

In the center of a square pond whose side measures 10 m grows a plant whose top reaches 1 m above the water level. If we pull the plant toward the bank, its top becomes even with the waters surface. What is the depth of the pond and the length of the plant?

Do not worry about your difficulties in mathematics. I can assure

you that mine are still greater.	Einstein

Equations and Inequalities

299

EXERCISES 4.5

1. Solve the inequality systems.

x+ 2 > 0

x 1> 0

2x 3 < 0

x 2

x2 + 3x 10 > 0

x+1

+11x 6 < 0

2x2

x2 1

> 0

x2 4x+ 3 < 0

2x2 + 2 < 5x

2x 4 < 0

x+ 2

< 0

x+1

> 2

x 1

< 0

x+1

2. Solve the inequality systems.

x+ 3

3 x < 2

a.x3 <16x b. 4x – 2 < x2 + 1 < 4x + 6

4 x2

c.	5x 7	< 4	x		3x
	x 5		5 x	+ x2 25		< 4

3.Solve the inequalities.

a.|4 – 3x| 2 – x

b.|x2 – 3x – 15| < 2x2 – x

c.|2x2 + x + 11| > x2 – 5x + 6

d.|x – 4| + |2x + 6| > 10

e.x2 5x+ 6 2

f.|2x + 1| – |5x – 2| 1

g.x2 x 12 2x x 3

4. Solve the inequalities.

a.	2x+10 < 3x 5	b.	x+ 3	2
			4 x
c.	x2 x 12 < x	d.	17 15x 2x3		> 0
			x+ 3
e.	x+1 x 2 1	f.	x 4	< x 8
e.	x+1 x 2 1	f.	x + 2	< x 8
			x + 2

5.Solve the inequalities.

a.x2 9 < x+ 2

b.2 3x x2 2x+ 3

c.x2 x 2 x 1

d.	4 1 x <			2 x
e.	x 2 + x 3 < 4
6. Solve the inequalities.
a.	2+ x	x	1	b. 1+ x	x	1

c.6 x2 < x 1

7.Solve the inequality



x+ 3 4 x 1 + x+8 6 x 1 >1.

8.For which values of m are the following inequalities

satisfied for any real number x?

a.2x2 + 5x + m > x2 – x – 7

b.2x2 + mx2 – 2x + m – 1 > (1 + m)x2 – 4x – m

c.(m + 2)x2 – 3x + m – 2 < 0

d.x2 – 6x + 4m2 > 7

9.Solve the inequality



(x2 + 6x + 14)2 – 9 (x2 + 6x + 15) + 9 < 0.

300	Algebra 10

CHAPTER REVIEW TEST 4A

1.Find the solutions of the quadratic equation 2x2 + 5x + 3 = 0.

A)	1	and 3			B)	3 and	1	C) 1 and		3
	2						2			2
	D)		1	and –3			E) 1 and		1
			2						2

2.–1 is a root of x2 + mx + n = 0 and m + n = 5. Find m.

A) 1

B) –1

C) 2

D) –2

E) 3

3.Which one of the following is false for the equation x2 – 7x + 1 = 0?

A)The roots are not integer.

B)The roots have the same sign.

C)The roots are rational.

D)The roots are both positive.

E)The sum of the roots is positive.

4.Find the discriminant of x2 – 2ñ5x + 4 = 0.

A) 2

B) 4

C) 36

D) 16

E) 24

5.The sum and product of the roots of the equation 2x2 – hx + 2k = 0 are 4 and –3, respectively.

Find the values of h and k.

A) 8 and –6	B) 4 and –3	C) –3 and 4
D) –3 and 8	E) 8 and –3

6.What is the sum of the roots of the equation x4 – 13x2 + 36 = 0?

A) 13

B) 18

C) 1

D) 0

E) –13

7.Which one of the following equations has roots –2 and 3?

A) 2x2 – x – 6 = 0	B) x2 + x – 6 = 0
C) x2 – x – 6 = 0	D) x2 + x + 6 = 0

E)2x2 + x – 6 = 0

8.The sum of a number and its reciprocal is 1225. Find the number.

A) 1	B) –1	C)	4	D) –	1	E) –3
			3		2

Chapter Review Test 4A

301

9. What are the roots of the equation x12 3x +2 = 0?

A) 1 and		1	B)	1	and 1	C)	1	and	4
	2			2			2		3
D) –		1	and –1			E) –1 and 1
		2

10.One of the roots of the equation ax2 + bx + c = 0 is twice the other root. What is the relation between the coefficients a, b, and c?


A) 4b2 = 4c	B) 2b2 = 9ac		C) 2b2 = 9a
D) b2 = 8ac		E) 9b2 = 2ac

11.The diagonal of a rectangle is 5 cm, and the area is 12 cm2. Find the perimeter of the rectangle.

A) 12

B) 24

C) 10

D) 14

E) 15

12.x – 1 x+1+1= 0. Find the value of 4x.

A) 5

B) 4

C) 0

D) 16

E) –4

13. What is the solution of the equation x = ñx + 6?

A) –9

B) 4

C) –4

D) 16

E) 9

14.Which one of the following is not a solution of the system

	5 x2 6 xy+ 5y2	= 29
		?
	7 x2 8 xy+7 y2	?
	7 x2 8 xy+7 y2	= 43

A) (2, 3)		B) (–2, –3)	C) (–2, 3)
	D) (3, 2)	E) (–3, –2)

15.Two technicians can complete a mailing job in 12 hours while working together. Alone, the first technician can complete the mailing job seven hours faster than the other technician. How long will it take each technician to complete the mailing alone?


A) 14 and 21	B) 21 and 28		C) 7 and 14
D) 11 and 18		E) 15 and 22

16. The equations 6x2 – 18x				+		12	=	0 and
mx2 – nx + 1 = 0			have the same solution. Find
m and n.
A) 3 and 1			B) 3 and	1		C)	3	and	1
2	2		2	2			2		2
D) 1	and	3	E)		3	and		1
2		2			2			2

302	Algebra 11

CHAPTER REVIEW TEST 4B

1.x1 and x2 are roots of the equation

x2 + (2 – x1)x + 7x2 = 0. What is x1 + x2?

A) –2

B) 5

C) 7

D) 10

E) –14

2.Which one of the following is true for the equation |x|2 + |x| – 6 = 0?

A)There is only one root.

B)The product of the roots is –3.

C)The product of the roots is 4.

D)The sum of the roots is 0.

E)The sum of the roots is 1.

3.The sum of the squares of the roots of the equation x2 + 2hx = 3 is 10. Find the value of h.

A)	1	B) –3 C)	3	D)	1	E) –1 or 1
	2		2		2

4. Find the discriminant of 2x2 – 2x + 5 = 0.

A) –36

B) 4

C) 25

D) –16

E) 9

5.The longest side of a right triangle is 3 cm less than twice the length of the medium side. The shortest side is 3 cm. Find the lengths of the other two sides.


A) 8 and 15	B) 7 and 14		C) 8 and 10
D) 4 and 5		E) 6 and 8

6.A triangle has area of 2 cm2. The base of the triangle is 3 cm longer than its height. Find the length of the base and the height of the triangle.

A) 5 and 2	B) 4 and 1	C) 6 and 3
D) 7 and 4		E) 8 and 5

7.Which one of the following equations has roots 1 – ñ3 and 1 + ñ3?

A) x2 – 2x – 2 = 0	B) x2 + 2x – 2 = 0
C) x2 – x – ñ3 = 0	D) x2 – 2x + 2 = 0

E)x2 – 6 = 0

8.The sum and product of two numbers are 21 and 104, respectively. What is the bigger number?

A) 16

B) 22

C) 13

D) 14

E) 19

Chapter Review Test 4B

303

9. What are the roots of the equation

		x	2			x
3			+7					– 6	= 0?
3			+7					– 6	= 0?
	x+1					x+1
A)		3	and 2				B)	1	and 2	C)	1	and	4
		4						2			2		3
		D)		1	and –3				E) –1 and 2
				2

10.Which number is a solution to the equation x2 + |x – 1| – 3 = 0?

A) 3	B)	2	C)	3	D) 0	E) –1
		2		2

11. x – y = 12 and x y = 108 are given. Find x + y. A) ±20 B) ±24 C) ±28 D) ±30 E) ±32

12. x – 5 =	x – 3 is given. Find the value of x.
A) 5	B) –7	C) 1	D) 7	E) –4

			1	1
13. What is the solution to the equation x2				+x4	= 2?
A) 9	B) 1	C) –4	D) 4	E) 16

14.Which one of the following is not a solution of the system

	x2	+y2 = 26
		?
	x2	?
	x2	y2 = 24

A) (1, –5)			B) (–5, –1)	C) (–5, 1)
		D) (5, 1)	E) (5, –1)

15.A pipe can fill a pool four hours faster than another pipe. The slower pipe filled the pool for seven hours, then the other pipe was opened. The pipes then filled the pool in two hours, working together. How long would each pipe take to fill the pool alone?

A) 14 and 18 hours	B) 8 and 12 hours
C) 11 and 15 hours	D) 9 and 13 hours

E)10 and 14 hours

16.The roots of the equation x3 – 3x + 7 = 0 are p, q, and r. What is the product (p + 1)(q + 1)(r + 1)?

A) –8

B) –9

C) 7

D) 8

E) 9

304	Algebra 11

CHAPTER REVIEW TEST 4C

1.What is the solution of the inequality 2 – 3(1 – x) < x + 1.8?

A) x > 1.4	B) x < 0.4	C) x > 0.7
D) x > –0.7	E) x < 1.4

2.What is the solution of the inequality x2 + 5x – 14 0?

A) [–2, 7]	B) [–7, 2]	C) (–7, 2)
D) [2, 7)	E) (–2, 7]

3. What is the solution of the inequality x2 4		> 0?
	x2 1
A) (– , –2) (1, )	B) (– , –2) (2, )
C) (– , –1) (1, )	D) – {–1, 1}

E)(– , –2) (–1, 1) (2, )

4.What is the solution of the inequality

(2 3x)(3x 3)						0?
	3x+1					0?
	3x+1
A)		1	,	2	[1,	)	B)	;				1		2
A)		3	,	3	[1,	)	B)	;				3			; 1
		3		3								3		3
C)	2		(1, + )						1	,	2		(1, + )
C)	3		(1, + )				D)		3	,	3		(1, + )
	3								3		3

E)	;	1	2
E)	;	3		; 1
		3	3

5. What is the solution of the inequality

E)–2 < x < 2

6.What is the greatest integer value of x which satisfies the inequality x2 – 13x + 36 < 0?

A) 13

B) 7

C) –8

D) 8

E) –13

7. What is the solution of the inequality

1	+	1	+	1	< 0?
x		x2		x3

A) (– , 0)					B) (0, )	C)	D) E) {0}

8. How many integer values of x are there which

satisfy the inequality \|x + 2\|			2x+7 ?
A) 1	B) 3	C) 4	D) 5	E) 7

Chapter Review Test 4C

305

9. What is the solution of the inequality

(1+ x2 )2001 (1 x)2002				0?
	2003	(3 x)	2005
	(x+2)
A) x > –2			B) x < –2		C) x < 3
	D) x > 3			E) –2 < x < 3

10.The equation 2x(ax – 4) – x2 + 6 = 0 has no real roots. What is the smallest possible integer value of a?

A) –1

B) 2

C) 3

D) 4

E) 5

11. What is the solution of the inequality

(x 3)(x+2) <1?
x2 1
A) (–2, –1) (1, 3)	B) (–2, –1) C) (1, 3)
D) (–5, –1) (1, )	E) (–1, 1) (5, )

12. What is the sum of the integer values of x which

satisfy the inequality			x2 8x+7		< 0?
		2
			(x+2)
A) 32	B) 28	C) 24		D) 20		E) 16

13. What is the domain of the function

y =	x 2	?

	20 8x x2
	20 8x x2
A) [–1, 7]		B) [–10, 2]	C) (–10, 2)
	D) [5, 4)	E) (–2, 10]

14. What is the solution of the inequality system

	x2 5x+6 0
		?
	x2 + x 2 0	?
	x2 + x 2 0

A) (2, 3)		B) [2, 3]	C) (–2, 3)
	D) [3, 2)	E) (–3, –2)

15. What is the solution of the inequality system

	x	< 2
	x	< 2
				?
			0	?
(x 3)(x 2)			0

A) (–2, 2)				B) [2, 3]	C) (–2, 3)
		D) [3, 2)		E) (–3, –2)

16. What is the solution of the inequality

3 x x+3 >1?
A) (–2, 2)	B) (1, )	C) (–2, –3)
D) (– , 2)		E) (2, )

306	Algebra 11

CHAPTER REVIEW TEST 4D

1.What is the solution of the inequality

x+12 x3 1 > x?

A) x > 1	B) x < –3	C) x < 3
D) x < 1		E) x > 6

2.What is the solution of the inequality (x – 6)(x + 3) 2 – 2x?

A) [–5, 4]	B) [–4, 5]	C) (–4, 2)
D) [–5, 4)	E) (–4, 5]

3. What is the solution of the inequality

x2 + x		<1?
5x 3		<1?
5x 3
	,		3	(1, 3)	B)	3		(3,	)
A)	,		5	(1, 3)	B)		, 1	(3,	)
			5			5
C) (– , 1)					D) – {–1, 1}

E)(–1, 1) (3, )

4.What is the solution of the inequality

1		+	5	<1?
	2 x		2+ x

A) [– , –1) (1, 3)					B) [3, ]
C) (– , –2) (2, )					D) (1, 3)
				E)	(1, 3)

5. What is the solution of the inequality

D) [2, )	E)	,	1
			2

6.What is the smallest possible integer value of x which satisfies the inequality (x + 7)2(x – 4) 0?

A) 3

B) 4

C) –8

D) 8

E) –13

7.What is the solution of the inequality (x2 + 8x – 9)(x2 – 4) 0?

A) (– , –2) [1, )	B) (– , –2] (2, )
C) (– , –1] [1, )	D) R – {–1, 1}

E)(– , –9] [–2, 1] [2, )

8.Which inequality is represented in this sign chart?

-2		5

+	–		+

A) x2 – 3x + 10 < 0		B) x2 > 3x – 10
C) x2 < 3x + 10		D) x2 – 2x + 5 0

E) x2 < 2x + 5

Chapter Review Test 4D

307

9. What is the solution of the inequality

(x 1)3(x 2)	0?
(x 3)2
A) (– , 1)	B) [2, ]	C) [1, 2]
D) [–1, 2)		E) (2, )

10.The equation 3x2 + bx + 12 = 0 has real roots. What are the possible values of b?


A) [–12, 12]	B) (– , –12]		C) [12,	)
D) (– , –12]	[12, )	E) (–1, 1) (5,		)

11. What is the solution of the inequality							x	1?
							x

							x+1
A)	1					B) (–2, –1)	C) (1, 3)
A)	2	,				B) (–2, –1)	C) (1, 3)
	2
	D)		1			E) (–1, 1)
	D)			2	,	E) (–1, 1)
				2

12. What is the sum of the integer values of x which


satisfy the inequality			(x 2)(x 4)x2		0?
satisfy the inequality			(x+1)2		0?
			(x+1)2
A) 8	B) 9	C) 10		D) 11	E) 16

13. What is the domain of the function

y = 4x x2 +	7		?
y = 4x x2 +			?
	x 2
A) (–2, 4]		B) [–1, 2]		C) (–4, 2)
D) [2, 4)			E) (2, 4]

14. What is the solution of the inequality system

	x 5	> 0
	x+ 3	> 0

			?
	x+7		?
	x+7	< 0
	x 3	< 0
	x 3
A) (2, 3)				B) [–7, 3]	C) (–7, –3)
	D) [–3, 3)				E) (7, –2)

15.How many integer values of x are there which satisfy the inequality (x + 8)(x - 1)2(x – 5) < 0?

A) 8

B) 9

C) 10

D) 11

E) 16

16. x1 and x2 are the roots of x2 – 3mx + m – 3 = 0.

1 + 1 > 4 is given. x1 x2

What are the possible values of m?

A) (– , )	B) [2, ]	C) (– , –3)
D) (3, 12)		E) (0, 12)

308	Algebra 11

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