Inequalities

A company is trying to determine the most profitable selling price for a new toy. Given that

• if the selling price of each toy is £30, the company expects to sell 1500 toys in one year

• if the selling price of each toy is £50, the company expects to sell 300 toys in one year

Using a linear model, with y being the expected number of toys sold in one year and x pounds being the selling price of the toy,

find an equation for y in terms of x. Given that

• the cost of making each toy is £10

• the company has additional costs of £8 000 per year

show that, according to the model, the yearly profit, P, in thousands of pounds, is given by

Use the model given in part (b) to answer parts (c), (d) and (e). Given that the company wishes to make a profit on sales of the toy,

find the range of possible selling prices of the toy.

Hence, or otherwise, deduce the selling price of the toy that maximises the profit. In one particular year, the company sold the toy for £35 and made £21 750 profit.

Use this information to evaluate the suitability of the model.

Figure 3 shows a sketch of the graph with equation y = f (x) where f (x) = 4 x – 3 – 5 x Given that a is a constant and a = 1

find the possible values of f (a) The function g is defined by g (x) = 2x + 17 x

Find the range of g f (x) The function h is defined by h (x) = kx x where k is a constant. Given that the equation f (x) = h (x) has no solutions,

find the range of values of k.

Figure 1 shows a sketch of the graph with equation

The vertex of the graph is at the point P, shown in Figure 1.

Find the coordinates of P.

Solve the equation

A line l has equation y = kx + 4 where k is a constant.

Given that l intersectsy 3 x 2 5at 2 distinct points,

find the range of values of k.

The circle C₁ has equation

Find

the coordinates of the centre of C₁

the radius of C₁

A different circle C₂

• has centre with coordinates (–6, –8)

• has radius k, where k is a constant

Given that C₁ and C₂ intersect at 2 distinct points,

find the range of values of k, writing your answer in set notation.

Figure 4 shows a sketch of the graph with equation y = | 2 x – 3 k | where k is a positive constant.

Sketch the graph with equation y = f (x) where f (x) = k – | 2 x – 3 k | stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of k, the set of values of x for which

giving your answer in set notation.

Find, in terms of k, the coordinates of the minimum point of the graph with equation

Figure 4 shows a sketch of the graph with equation

y = | 2 x – 3 k |

where k is a positive constant.

Sketch the graph with equation y = f (x) where

f (x) = k – | 2 x – 3 k |

stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of k, the set of values of x for which

giving your answer in set notation.

Find, in terms of k, the coordinates of the minimum point of the graph with equation

Figure 1 shows a sketch of a curve C with equation y = f (x) and a straight line l. The curve C meets l at the points (−2, 13) and (0, 25) as shown. The shaded region R is bounded by C and l as shown in Figure 1. Given that ● f (x) is a quadratic function in x ● (−2, 13) is the minimum turning point of y = f (x) use inequalities to define R.

The function g is defined by

where k is a constant.

Deduce the value of k.

Prove that

for all values of x in the domain of g.

Find the range of values of a for which

Figure 1 shows the plan of a garden. The marked angles are right angles. The six edges are straight lines. The lengths shown in the diagram are given in metres. Given that the perimeter of the garden is greater than 40 m,

show that x 1.7

Given that the area of the garden is less than 120 m²,

form and solve a quadratic inequality in x.

Hence state the range of the possible values of x.

Find the set of values of x for which

2(3 x + 4) > 1 – x

Arectangular room has a width ofx m.

The length of the room is 4 m longer than its width. Given that the perimeter of the room is greater than 19.2 m,

Given also that the area of the room is less than 21 m2,

write down an inequality, in terms ofx, for the area of the room.

Solve this inequality. (4)

Hence find the range of possible values for x. (1)

The equation

has two distinct real solutions for x.

Show that k satisfies

Hence find the set of possible values of k.

Find the set of values of x for which

3(x – 2) < 8 – 2 x

(2 x – 7)(1 + x) < 0

both 3(x – 2) < 8 – 2 x and (2 x – 7)(1 + x) < 0

Find the set of values of x for which

4 x – 3 > 7 – x

both 4x – 3 > 7 – x and 2x² – 5x – 12 < 0

The equation kx² + 4x + (5 – k) = 0, where k is a constant, has 2 different real solutions for x.

Show that k satisfies

Hence find the set of possible values of k.

The equation

has no real solutions for x.

Hence find the set of possible values of k.