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1.03 Solving linear inequalities graphically

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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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 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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5

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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6

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.

7

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.

8

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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9

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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10

Consider the function y = 2x - 6.

a

Sketch the graph of the line on a number plane.

b

Hence, solve the inequality 2x - 6 \gt 0.

11

Consider the function y = \dfrac{1}{2}x + 5.

a

Sketch the graph of the line on a number plane.

b

Hence, solve the inequality \dfrac{1}{2}x + 5\lt0

12

Consider the functions y = 2\left(x+1\right) \text{ and } y = - 3

a

Sketch both lines on the same number plane.

b

Hence, solve the inequality 2\left(x+1\right) \gt - 3

13

Consider the functions y = x + 4 \text{ and } y = 2x - 1

a

Sketch both lines on the same number plane.

b

Hence, solve the inequality x+4\lt2x-1

Equivalent inequalities
14

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?

15

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?

16

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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17

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

Using the graphs, solve the inequality

- 2 x - 6 > 0.

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a

Consider the functions y = 2x+1 \text{ and } y = 4 - x.

i

Sketch both lines on the same number plane.

ii

Hence, solve the inequality 2x + 1 \lt 4 - x.

b

Consider the functions y = 3x-3 \text{ and } y = 0.

i

Sketch both lines on the same number plane.

ii

Hence, solve the inequality 3x - 3\lt 0.

c

Explain why the solutions to the above inequalities are the same.

19

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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MA12-1

uses detailed algebraic and graphical techniques to critically construct, model and evaluate arguments in a range of familiar and unfamiliar contexts

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