Download Inequalities - Lesson 1-5 ANSWERS

Math 9
Inequalities
Lesson 1-5
Name SOLUTIONS
1. When the inequation is solved for m, does the inequality sign reverse direction?
Explain.
m
 5
a) 3m > -16
b)
c)  m  12
2
No – divided by a positive
No – Multiplied by a
Yes – Divided by a negative
positive
2. List in order the steps used to solve the inequation.
a) 4d – 3 > 6
b) -5 – 3a  8
c) 9t – 3  6t + 11
Add 3 to both sides
Divide by 4 on each side
Add 5 to both sides
Divide by -3 to both sides
Switch direction of sign
Add 3 to both sides
Subtract 6t from both sides
Divide by 3 to both sides
3. Solve, then check.
a) 2n + 3 < 5
2n + 3 – 3  5 – 3
2n  2
2n 2

2 2
n 1
-6 + 5 < - 5 + 5 +
1 
n
2
n
2( 1)  2 
2
2n
n
2
b) -4 – 3r > 0
c) -6 < -5 +
n
2
d) 3(b – 5) < 6b – 3
-4 + 4 – 3r > 0 + 4
- 3r > 4
 3r
4

3 3
1
r  1
3
3b – 15 < 6b – 3
3b – 3b – 15 < 6b – 3b – 3
-15 < 3b – 3
-15 + 3 < 3b – 3 + 3
-12 < 3b
 12 3b

3
3
4b
4. Twice Elva’s age is greater than or equal to 76.
a) Write the inequation.
b) Solve the inequation, then check
c) At least how old must Elva be?
2a  76
2a  76
2a 76

2
2
a  38
She must be at least 38
years old.
5. Determine if the given value for n makes a true statement.
n
n
 2 , n = -11 c) 2n + 3 > 9, n = 3 d)  6, n  9
a) n  7, n = 4 b)
5
2
(4 )  7
 n 
 5
  5(2)
Makes a true statement
 5
n  10
(11)  10
Makes a false statement
2n + 3 > 9
n
2   2(6)
2n + 3 – 3 > 9 – 3
2
2n > 6
n  12
n>3
(9)  12
(3) > 3
Makes a false statement
Makes a true statement
6. Solve, then check.
a) -5n + 1 < 16
b) 2b + 3b < 15
e) 8z – 2 > 6z + 12 f) 6f – 8f  6
n
5 1
i)
j) -3w + 4 < 16
3
-5n + 1 < 16
2b + 3b < 15
-5n + 1 – 1 < 16 – 1
5b < 15
-5n < 15
5b 15

 5n 15
5
5

5 5
b3
n  3
c3
 4
2
 c  3
 2
  2( 4)
 2 
c38
c 33  83
c5
c) 9y + 7  - 11
g) -3(x + 4)  6
8z – 2 > 6z + 12
8z – 2 + 2 > 6z + 12 + 2
8z > 6z + 14
8z – 6z > 6z – 6z + 14
2z > 14
z>7
c3
 4
2
h) 10t – 3 > 4t + 15
d)
9y + 7  - 11
9y + 7 – 7  - 11 – 7
9y  - 18
9 y  18

9
9
y  2
6f – 8f  6
-2f  6
2f
6

2
2
f  3
-3(x + 4)  6
-3x – 12  6
-3x – 12 + 12  6 + 12
-3x  18
 3 x 18

3 3
x  6
10t – 3 > 4t + 15
10t – 3 + 3 > 4t + 15 + 3
10t > 4t + 18
10t – 4t > 4t – 4t + 18
6t > 18
t>3
n
5 1
3
n
3   3(5)  3(1)
3
n  15  3
n  15  15  3  15
n  12
-3w + 4 < 16
-3w + 4 – 4 < 16 – 4
-3w < 12
 3w 12

3
3
w  4
7. Solve.
a) 3x + 7  19 b) 12 – 3x > 0
c) 5x – 3 < 15
d) 8 + 7x  13
12
–
12
–
3x
>
0
12
5x
–
3
+
3
<
15
+
3
3x + 7 – 7  19 – 7
8 – 8 + 7x  13 – 8
-3x > 0
5x < 18
3x  12
7x  5

3
x

12
18
5
x4

x<
x
3
3
5
7
x < 3.6
x4
8. Solve.
a) 3x + 2x – 7 > 8
5x – 7 + 7 > 8 + 7
5x > 15
x>3
b) 7x – 4x + 6  18
3x + 6  18
3x + 6 – 6  18 – 6
3x  12
x4
c) x – 5x + 1  9
-4x + 1  9
-4x + 1 – 1  9 – 1
- 4x  8
 4x
8

4 4
x  2
9. Solve.
a) 5x – 2 < 3x – 10 b) 2x + 7  8x – 11 c) 4x – 9 > 7x + 24
d) 3x – x + 5 < 1
2x + 5 <1
2x + 5 – 5 < 1 – 5
2x < - 4
x<-2
d) 8 – 3x  17 – x
e) 3x + 19  8x – 12 f) 6x – 12x + 5  3
5x – 2 + 2 < 3x – 10 + 2
5x < 3x – 8
5x – 3x < 3x – 3x – 8
2x < - 8
x<-4
2x + 7 + 11  8x – 11 + 11
2x + 18  8x
2x – 2x + 18  8x – 2x
18  6x
3x
4x – 9 – 24 > 7x + 24 – 24
4x – 33 > 7x
4x – 4x – 33 > 7x – 4x
-33 > 3x
-11 > x
8 – 17 – 3x  17 – 17 – x
-9 – 3x  - x
- 9 – 3x + 3x  - x + 3x
- 9  2x
-4.5  x
3x + 19 + 12  8x – 12 + 12
3x + 31  8x
3x – 3x + 31  8x – 3x
31  5x
6.2  x
-6x + 5  3
- 6x + 5 – 5  3 – 5
- 6x  - 2
 6x  2

6
6
1
x
3
10. Solve
a) 2(x + 8) > 4(3 + x) b) 9x – 3  3(x – 4) c) 3(1 – x)  -2(2 – x)
1
2
d) (2  5 x)  (15  3 x)
e) -9(x + 3) – 9x < - 3x – (3 – x) + 8
2
3
2x + 16 > 12 + 4x
9x – 3  3x – 12
3 – 3x  - 4 + 2x
2x + 16 – 12 > 12 – 12 + 4x
9x – 3 + 3  3x – 12 + 3
3 + 4 – 3x  - 4 + 4 + 2x
2x + 4 > 4x
9x  3x – 9
7 – 3x  2x
2x – 2x + 4 > 4x – 2x
9x – 3x  3x – 3x – 9
7 – 3x + 3x  2x + 3x
+ 4 > 2x
6x  - 9
7  5x
2>x
x  -1.5
1.4  x
-9x
–
27
–
9x
<
3x
–
3
+
x
1

2

6  2  5 x   6  15  3 x  + 8
2

3

-18x – 27 < - 2x + 5
3(2  5 x)  4(15  3 x)
- 18x + 18x – 27 < - 2x +
6  15 x  60  12 x
18x + 5
- 27 < 16x + 5
6  6  15 x  60  6  12 x
-27 – 5 < 16x + 5 – 5
15 x  54  12 x
-32 < 16x
15 x  12 x  54  12 x  12 x
-2 < x
27 x  54
x2
11. Express as an inequation.
a) Three times a number, reduced b 8, is less than 10.
b) Four more than twice a number is greater than 2.
c) If the load is increased by 500 kg, the vehicle will carry more than 2500
kg.
3x – 8 < 10
2x + 4 > 2
X + 500 > 2500
12. Bernard is four years older than Allan. At least how old must each be if the sum
of their ages is greater than 26?
Let x represent Allen’s age
Let x + 4 represent Bernard’s age
Allen must be at least 12
Bernard must be at least 16
Inequation is: x + (x + 4) > 26
2x + 4 > 26
2x + 4 – 4 > 26 – 4
2x > 22
x > 11
13. When an integer is doubled and then increased by 5, the result is less than 11.
a) Find three possible integers for which this statement is true.
b) Find three possible integers for which it is false.
2x + 5 < 11
2x + 5 – 5 < 11 – 5
2x < 6
x<3
2x + 5 < 11
2,1,0
4,5,6
14. a) Write an expression for the perimeter of the regular hexagon.
b) The perimeter is less than 48 cm. Express this as an inequation.
c) Solve the inequation, and list three possible lengths for the sides.
6(x+1) or 6x + 6
6x + 6 < 48
6x + 6 < 48
6x < 48 – 6
6x < 42
x<7
Possible answers are = 6,5,4
15. Wanda earns $19.50/h as an assistant librarian. What is the least number of hours
that she works to earn more than $660.00?
19.5h > 660
She has to work more
19.5h 660

than 33.85 hours.
19.5 19.5
h  33.85