iGCSE (2021 Edition)

# 4.08 Graphical solutions of equations and 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.

2

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

3

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

4

Consider the graphs of y = x + 6 and

y = x - 7:

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

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.

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.

9

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

Using the graphs, solve the inequality

x + 5 > 12 - x.

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

14

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
15

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)?

16

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

a

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

b

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

c

How many real solutions are there to f \left( x \right) = 0?

17

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.

18

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)?

19

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.

20

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.

21

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.

22

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.

23

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.

24

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.

25

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.

26

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.

27

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
28

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?

29

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?

30

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

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

31

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

Using the graphs, solve the inequality

- 2 x - 6 > 0.

32
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.

33

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

### Outcomes

#### 0606C1.3

Understand the relationship between y = f(x) and y = |f(x)|, where f(x) may be linear, quadratic or trigonometric.

#### 0606C3.4

Sketch the graphs of cubic polynomials and their moduli, when given in factorised form y = k(x – a)(x – b)(x – c).