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P R AC T I C E T E S T, p a g e s 4 1 5 – 4 1 8
1. Multiple Choice Which inequality is not represented by this graph?
A. y > x2 - x - 6
2
B. y > ax - 2b - 4
1
y
2
25
x
2
C. y > (x + 2)(x - 3)
0
2
2
D. y > (x + 3)(x - 2)
4
6
2. Multiple Choice Which inequality below is represented by this
number line?
⫺6
⫺5
⫺4
⫺3
⫺2
⫺1
0
1
2
1
A. 2x2 + 7x - 4 ≥ 0
B. 2x2 + 7x - 4 ≤ 0
C. -2x2 - 7x + 4 ≥ 0
D. 2x2 - 7x + 4 ≤ 0
2
3. Graph each inequality. Give 2 possible solutions in each case.
a) 2x2 - 5x < -2
Solve: 2x2 ⴚ 5x ⴙ 2 ⴝ 0
(2x ⴚ 1)(x ⴚ 2) ⴝ 0
x ⴝ 0.5 or x ⴝ 2
When x<0.5, such as x ⴝ 0, L.S. ⴝ 0; R.S. ⴝ ⴚ2;
so x ⴝ 0 does not satisfy the inequality.
When 0.5<x<2, such as x ⴝ 1, L.S. ⴝ ⴚ3; R.S. ⴝ ⴚ2;
so x ⴝ 1 does satisfy the inequality.
The solution is: 0.5<x<2, x ç ⺢
Two possible solutions are: x ⴝ 1 and x ⴝ 1.5
⫺1
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b) -2 ≥ -0.5(x - 6)2
Solve: ⴚ2 ⴝ ⴚ0.5(x ⴚ 6)2
(x ⴚ 6)2 ⴝ 4
x ⴚ 6 ⴝ —2
x ⴝ 4 or x ⴝ 8
When x ◊ 4, such as x ⴝ 0, L.S. ⴝ ⴚ2; R.S. ⴝ ⴚ18;
so x ⴝ 0 does satisfy the inequality.
When x » 8, such as x ⴝ 10, L.S. ⴝ ⴚ2; R.S. ⴝ ⴚ8;
so x ⴝ 10 does satisfy the inequality.
The solution is: x ◊ 4 or x » 8, x ç ⺢
Two possible solutions are: x ⴝ 1 and x ⴝ 20
3
4
5
6
7
Graph the related functions.
The line has slope 0.5 and
y-intercept ⴚ2.
Draw a solid line. Shade
the region below the line.
Graph of y 0.5x 2
y
x
x
y
4
2
4
9
10
d) y > - (x + 3)2 + 4
c) y ≤ 0.5x - 2
0
8
The parabola is congruent to
y ⴝ ⴚx2 and has vertex (ⴚ3, 4).
Draw a broken curve. Shade
the region above the curve.
Graph of y (x 3)2 4
y
4
6
2
.5
0
x
6
4
2
0
6
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4. At a school cafeteria, an apple costs 75¢ and a banana costs 50¢. Ava
has up to $5 to spend on fruit for herself and her friends.
a) Write an inequality to represent this situation. What are the
restrictions on the variables?
Let a represent the number of apples
and b represent the number of bananas.
An inequality is: 75a ⴙ 50b ◊ 500, or
3a ⴙ 2b ◊ 20
Both a and b are whole numbers.
b) Determine 2 possible ways that Ava can spend up to $5.
b
12
2b
4
8
3a
20
Determine the coordinates of 2 points
that satisfy the related function.
When a ⴝ 0, b ⴝ 10
When a ⴝ 6, b ⴝ 1
Join the points with a solid line.
The solution is the points, with whole-number
coordinates, on and below the line.
Two ways are: 4 apples, 2 bananas; 2 apples, 6 bananas
0
4
a
8
12
5. Solve each system of equations. Use algebra for one system and
graphing technology for the other. How did you decide which
strategy to use?
a) y = 2x2 + x - 1 ➀
x + y = 12
➁
Rearrange equation ➁.
y ⴝ 12 ⴚ x
Substitute y ⴝ 12 ⴚ x in equation ➀.
12 ⴚ x ⴝ 2x2 ⴙ x ⴚ 1
2
2x ⴙ 2x ⴚ 13 ⴝ 0
This equation does not factor, so I use graphing technology. I use
algebra when the equation does factor.
Input the equations. To the nearest tenth, the graphs intersect at these
points: (ⴚ3.1, 15.1) and (2.1, 9.9)
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b) y = (x - 2)2
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➀
y = -x2 + 4x - 4 ➁
From equation ➀, substitute y ⴝ (x ⴚ 2)2 in equation ➁.
(x ⴚ 2)2 ⴝ ⴚx2 ⴙ 4x ⴚ 4
x2 ⴚ 4x ⴙ 4 ⴝ ⴚx2 ⴙ 4x ⴚ 4
2x2 ⴚ 8x ⴙ 8 ⴝ 0
x2 ⴚ 4x ⴙ 4 ⴝ 0
(x ⴚ 2)2 ⴝ 0
So, x ⴝ 2
Substitute x ⴝ 2 in equation ➀.
y ⴝ (2 ⴚ 2)2
yⴝ0
The solution is: (2, 0)
6. The cross section of a pedestrian tunnel under a road is parabolic
and is modelled by the equation y = - 0.3x2 + 1.8x, where
y metres is the height of the tunnel at a distance of x metres
measured horizontally from one edge of the path under the tunnel.
In 2010, the tallest living person was about 2.56 m tall. Could he
walk through the tunnel without having to bend over?
How could you use an inequality to solve this problem?
Determine the values of x for which y » 2.56.
Solve: ⴚ 0.3x2 ⴙ 1.8x » 2.56, or ⴚ0.3x2 ⴙ 1.8x ⴚ 2.56 » 0
Use graphing technology. Input: y ⴝ ⴚ0.3x2 ⴙ 1.8x ⴚ 2.56
Determine if there are any values of x for which y » 0; these values are
approximately 2.3 ◊ x ◊ 3.7.
So, the tallest living person could walk through the tunnel.
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