25. Graphical Inequalities - Cardiff Tutor Company

Question 1

Which of these graphs gives region $R$ , bounded by the following lines.

y\leq 4\\

y \geq 2\\

x\geq -3\\

x\leq 1\\

ANSWER: Multiple Choice (Type 1)

Answer: A

Workings:

Using your own graph, draw each inequality onto it, but making each equation equal to instead, e.g. $y=4$ .

Because we are only dealing with inclusive inequalities all lines should be solid.

Once all four lines have been drawn on, label the region bounded by them as $R$ .

Select the graph that matches your own.

Marks = 2

Question 2

Which of these graphs gives region $R$ , bounded by the following lines.

y > 4x-3

y < 6

x > 0

ANSWER: Multiple Choice (Type 1)

Answer: C

Workings:

Using your own graph, draw the inequalities onto it, swapping the inequality symbols for equals, e.g. $y=4x-3$ .

For this question we are only dealing with strict inequalities so all lines should be dashed.

Label the region bounded by the lines as $R$ .

Select the graph that matches your own.

Marks = 3

Question 3

Which of these graphs gives region R, bounded by the following lines.

y\leq x+1\\

y > -3\\

x\leq 4

ANSWER: Multiple Choice (Type 1)

Answer: B

Workings:

Using your own graph, draw the inequalities onto it, swapping the inequality symbols for equals, e.g. $y=x+1$ .

$y=x+1$ and $x=4$ are inclusive inequalities so should be solid while $y=-3$ is $>$ so should be dashed.

Label the region bounded by the lines as $R$ .

Select the graph that matches your own.

Marks = 3

Question 4

Which of these graphs gives region $R$ , bounded by the inequality

x+3y\leq -12

ANSWER: Multiple Choice (Type 1)

Answer: B

Workings:

The inequality can be rearranged to $y\leq -4-\dfrac{x}{3}$

Using your own graph, draw the inequality onto it, swapping the inequality symbols for equals, i.e. $y=-4-\dfrac{x}{3}$ .

Because $y\leq -4-\dfrac{x}{3}$ features a $\leq$ symbol the line should be solid.

Label the region bounded by the lines as $R$ .

Select the graph that matches your own.

Marks = 3

Question 5

Below is a graph showing the shaded region $A$ .

Find the three inequalities which satisfy the shaded region A.

ANSWER: Multiple Choice (Type 1)

A: $y\leq 2$ , $y\geq -x$ , $x\geq 4$

B: $y\leq -4$ , $y\geq x$ , $x\geq 1$

C: $y\leq 4$ , $y\geq x$ , $x\geq 4$

D: $y\leq 2$ , $y\geq 2x$ , $x\geq 0$

Answer: C

Workings:

There is a horizontal straight line intersecting the $y$ -axis at $y=4$ and a vertical line intersecting the $x$ -axis at $x=-4$ .

The area below $y=4$ is shaded with a solid line so the inequality must be $y\leq 4$ .

The area to the right of $x=-4$ is shaded with a solid line so the inequality must be $x\geq -4$ .

We also have a line with gradient $= 1$ intersecting the origin, so we have the line $y=x$ .

The area above $y=x$ is shaded with a solid line so the inequality must be $y\geq x$ .

Marks = 3

Question 6

Below is a graph showing the shaded region $B$ .

Find the three inequalities which satisfy the shaded region $B$ .

ANSWER: Multiple Choice (Type 1)

A: $y\geq 2x-4$ , $x > 3$ , $y\leq \dfrac{x}{3}+1$

B: $y\geq x-5$ , $x > 2$ , $y\leq \dfrac{x}{4}+5$

C: $y\geq 2x-6$ , $x > 0$ , $y\leq \dfrac{x}{2}+3$

D: $y\geq 2x-6$ , $x > 2$ , $y\leq \dfrac{x}{4}+2$

Answer: D

Workings:

The first line intersects the $y$ -axis at $-6$ and has a gradient of $2$ . This gives an equation of $y=2x-6$ .

The area above the line is shaded with a solid line, so the inequality must be $y\geq 2x-6$ .

The second line intersects the $y$ -axis at $2$ and has a gradient of $\dfrac{1}{4}$ . This gives an equation of $y=\dfrac{1}{4}x+2$ .

The area below the line is shaded with a dashed line, so the inequality must be $y<\dfrac{1}{4}x+2$ .

The third line is a vertical line intersecting the $x$ -axis at $x=2$ .

The area to the right of the line is shaded with a dashed line, so the inequality must be $x > 2$ .

Marks = 3