Inequalities (Edexcel A Level Maths: Pure): Exam Questions

In this question you should show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Using algebra, solve the inequality

writing your answer in set notation.

Using algebra, solve the inequality

On the axes below sketch the region defined by the inequalities

Label your region .

Given that

and that

find the possible values of .

The cross section of a tunnel is in the shape of the region defined by the inequalities

where and are measured in metres.

Sketch the region on the axes below and label it .

Write down the maximum height and maximum width of the tunnel.

The equation

where  is a constant, has no real solutions.

Find the possible value(s) of .

A stone is fired vertically upwards from ground level.

The vertical distance above the ground, metres at time  seconds after launch, is given by

Find the length of time that the stone spends at a height greater than metres above the ground.

Give your answer to 3 significant figures.

On the axes below sketch the region defined by the inequalities

Label your region .

Find the three inequalities that define the region shown in the diagram below.

Given that and are integers such that

show that

Figure 3 shows a sketch of a curve and a straight line .

Given that

• has equation where is a quadratic expression in

• cuts the -axis at 0 and 6

• cuts the -axis at 60 and intersects at the point

use inequalities to define the region shown shaded in Figure 3.

The equation

where  is a constant has two distinct real roots.

Find the possible values of .

Using algebra, solve the inequality

Use algebra to solve the inequality

The equation

where  is a constant, has two distinct real roots. 

Find the possible values of .

Find the values of  that satisfy both of the inequalities below.

Give your answer in set notation.

An electronics company produces cables.

The company sells cables individually for  pence each.

Write down an expression, in terms of , for the total income (in pence) made from selling cables.

The cost to the company of making cables is  pence.

By forming and solving a suitable inequality, find the minimum number of cables the company must sell in order to make a profit.

Find the values of  that satisfy both of the inequalities below.

Use algebra to solve the inequality

On the axes below sketch the region defined by the inequalities

Label this region .

Use algebra to solve the inequality

writing your answer in set notation.

Figure 4 shows a sketch of the graph of , where

Find all the values of for which

Figure 1 shows a sketch of a curve with equation and a straight line .

The curve meets at the points (-2, 13) and (0, 25) as shown.

The shaded region is bounded by and as shown in Figure 1.

Given that

• is a quadratic function in

• (-2, 13) is the minimum turning point of

use inequalities to define .

The equation 

has real roots.

Find the possible values of .

Use algebra to solve the inequality

writing your answer in interval notation.

Find the three inequalities that define the region shown in the diagram below.

A stone is projected vertically upwards from an initial height of  metres above the ground.

• The vertical height of the stone above its initial position, metres, at time  seconds after launch, is given by

At the same time, a second stone is projected vertically upwards from an height of metres above the ground.

• The vertical height of the second stone above its initial position, metres, at time  seconds after launch, is given by

Find the length of time during which both stones are greater than metres above the ground.

Give your answer to 3 significant figures.

A company produces chairs and tables in a day. 

They sell every chair and every table they produce. 

Due to the manufacturing processes involved, the number of chairs and tables they can make in a day are limited by the following five inequalities:

Explain the significance of the inequalities  and  in the context.

On the axes below, sketch the region within which the company can produce  chairs and tables per day.

The company’s profit, per day, is given by the formula

A financial advisor tells the company that their maximum profit is found when and lie on a vertex of the region found in part (b), but did not say which vertex.

Use this information to find the number of chairs and tables the company should make in order to maximise its daily profit.

Show your working clearly.