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Middle Years

4.11 Non-linear inequalities

Worksheet
Linear equations and inequalities
1

Consider the graph of y = 2 x + 6. Using the graph, solve the inequality 2 x + 6 \geq 0.

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2

Consider the graph of y = x - 6. Using the graph, solve the inequality x - 6 < 0.

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Consider the graph of y = 5 x + 3. Using the graph, solve the inequality 5 x + 3 \leq 0.

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4

Consider the equation 2 \left(x - 1\right) - 3 = 7.

a

Solve for the value of x that satisfies the equation.

b

To verify the solution graphically, what two straight lines need to be graphed?

c

Graph these lines on the same number plane.

d

Hence find the value of x that satisfies the two equations.

5

Consider the graphs of y = x + 6 and

y = x - 7:

How many solutions does the inequality x + 6 \geq x - 7 have?

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6

Consider the inequality 2 x - 4 > 2 - 4 x.

a

Sketch the graphs of the lines for y = 2 x - 4 and y = 2 - 4 x.

b

Find the point of intersection of the lines.

c

Hence, solve the inequality 2 x - 4 > 2 - 4 x.

7

Consider the graph of the lines y = 17 and\\ y = 4 x - 3:

Using the graphs, solve the inequality

4 x - 3 < 17.

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8

Consider the graphs of y = x + 5 and \\ y = 12 - x:

Using the graphs, solve the inequality

x + 5 > 12 - x.

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9

Use sign diagrams to solve the following inequalities:

a
5x-15 \gt 0
b
8x+32 \leq 0
c
-3x+81 \geq 0
d
-20x \gt 80
Quadratic equations and inequalities
10

Consider the graph of y = f \left( x \right):

a

Find the values of x for which f \left( x \right) = 0.

b

For what values of x is f \left( x \right) < 0?

c

For what values of x is f \left( x \right) > 0?

d

What is the x-coordinate of the vertex of f \left( x \right)?

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11

Consider the function f \left( x \right) = 5 + 4 x - x^{2}:

Use the graph to solve the inequality

5 + 4 x - x^{2} > 0.

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12

Consider the graph of y = f \left( x \right):

a

For what values of x is f \left( x \right) < 0?

b

For what value of x is f \left( x \right) \geq 0?

c

What is the axis of symmetry of f \left( x \right)?

d

What is the value of the discriminant of f \left( x \right)?

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13

Consider the function f \left( x \right) = x^{2} - 4 x - 5.

a

Sketch the graph of the function.

b

Hence state the values of x for which f \left( x \right) \leq 0.

14

Consider the function f \left( x \right) = 3 x^{2} - 2 x - 8.

a

Solve the equation f \left( x \right) = 0.

b

Sketch the graph of the function.

c

Hence state the values of x for which f \left( x \right) \geq 0.

15

Consider the inequality \left(x - 3\right)^{2} \leq 0.

a

How many x-intercepts does the graph of y = \left(x - 3\right)^{2} have?

b

Solve the inequality.

16

Consider the function y = 2 x^{2} + 9 x + 8.

a

Determine the x-intercepts of the function.

b

Is the graph concave up or concave down?

c

Hence find the values of x for which y > 0.

17

Consider the function f \left( x \right) = x^{2} - 2 x.

a

Sketch the graph of the function.

b

Hence state the values of x for which f \left( x \right) \leq 8.

18

Consider the inequality x^{2} - 2 x \leq - x + 2.

a

Sketch the graphs of y = x^{2} - 2 x and y = - x + 2 on the same number plane.

b

State the x-values for the points of intersection.

c

Hence solve the inequality x^{2} - 2 x \leq - x + 2.

19

Consider the inequality x^{2} > 6 x - 5.

a

Sketch the graphs of y = x^{2} and y = 6 x - 5 on the same number plane.

b

State the x-values for the points of intersection.

c

Hence solve the inequality x^{2} > 6 x - 5.

20

Consider the inequality 3 x^{2} + x \geq 2 x^{2} + 2.

a

Sketch the graphs of y = 3 x^{2} + x and y = 2 x^{2} + 2 on the same number plane.

b

Hence or otherwise, solve the inequality 3 x^{2} + x \geq 2 x^{2} + 2.

21

Use sign diagrams to solve the following inequalities:

a
x^2 - 9 \gt 0
b
2x^2 -4x \leq 0
c
x^2 - 5x + 6 \geq 0
d
x^2 +7x +12 \gt 0
Equivalent inequalities
22

To solve the inequality x \leq 2 x - 3, Christa graphed y = x + 3. What other line would she need to graph to be able to solve the inequality graphically?

23

To solve the inequality x \leq \dfrac{x - 3}{4} - 1, Tracy graphed y = x - 3. What other line would she need to graph to be able to solve the inequality graphically?

24

Consider the graph of y = - \dfrac{2}{3} x - 2.

Using the graphs, solve the inequality

- 2 x - 6 > 0.

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25

Consider the graph of the lines y = 3 and

y = 23 - 4 x:

a

Using the graphs, solve the inequality

23 - 4 x < 3.

b

Using the graphs, solve the inequality - 20 + 4 x \geq 0.

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26

Consider the graphs of y = 3 x + 4 and \\ y = x:

Using the graphs, solve the inequality \\ 3 x + 4 - x \leq 0.

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27

Explain whether the following inequalities can be solved graphically using the graphs of: f(x) = 4 x + 3 \text{ and }g(x) = 5 - x

a

\left( 4 x + 3\right) - \left(5 - x\right) \leq 0

b

\left( 4 x + 3\right) + \left(3 - x\right) < 0

c

\dfrac{4 x + 3}{5} + x > 0

d

3 x + 8 < 0

e

5 x > 2

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