(a)(i) Solve $-4<-\frac{1}{2}x$.

(ii) Show the solution on the number line.

The shaded region is defined by four inequalities.
Two of these inequalities are $x\ge 0$ and $y\le 5-\frac{5}{7}x$. (a) Write the remaining two inequalities.

(b) Explain why the function 14y + 10x attains its maximum at multiple points rather than at one point in the region.

(a) It is given that:
$72\le q\le 88$
$14\le r\le 16$
$100\le m\le 150$
Find the maximum possible value of $\frac{r}{m-q}$, leaving your answer as a fraction in its simplest form.

(a) The inequality $16<9-5x<24$ has solution $a<x<b$.
Find the values of a and b.

Find the integer values of x for which $1-x<3x+5\le x+9$.

Find the integer values for n which satisfy this inequality:
$-3<2n\le 5$.

Given that $-4\le t\le 2$ and $5\le r\le 7$,
Find the greatest possible value of:
(a) $r-t$,

(b) $\frac{3t^2}{r}$.

A carpenter makes x tables and y chairs per week.
The total number of tables and chairs made cannot exceed 16.
(a) Two of the inequalities are x ≥ 0 and y ≥ 0.
Write the third inequality in x and y to represent the information.

(b) The carpenter makes a profit of M20 per table and M30 per chair on his sales.
The total profit made cannot be less than M360.
Show that the inequality representing this information is $2x+3y\ge 36$.

(c) By shading the unwanted regions, show all this information on the grid.

(d) Find the:
(i) Maximum profit that the carpenter can make,

(ii) Number of tables and chairs that can be made in order to get a maximum profit.

A school transports learners to a stadium using two types of vehicles (mini-buses and taxis). Each trip should have at least one mini-bus and one taxi but not more than six vehicles altogether.
A mini-bus can carry up to 60 learners while a taxi can carry up to 15 learners.
(a) (i) Given that every vehicle must carry the maximum number of learners, find the minimum number of learners that can be transported in one trip.

(ii) Taking b for the number of mini-buses used and t for the number of taxis used, write an expression for the number of learners that can be transported.

(b) The school has at most 180 learners to transport.
(i) Use this information to form an inequality in b and t.

(ii) Two other inequalities defining region R, of the possible values of b and t are t ≥ 1, b ≥ 1 and b + t ≤ 6.
Use the inequalities to represent the region R on the grid provided.

(c) Transport costs are M800 per mini-bus and M400 per taxi.
Find the least possible cost, to the school, of transporting the learners.

A factory makes two types of jeans, type A and type B.
Each month, x of type A and y of type B jeans are made.
The following constraints control the daily production:

The diagram shows the four constraints.

One of the constraints is x + y ≤ 80.
(a) Write down in terms of x and/or y the other three constraints.

(b) On the diagram, shade the region that satisfies all the constraints.

(c) $y=-2x+\frac{P}{150}$ is the function that represents the profit, P, in Maloti.
Find the profit of each type of pair of jeans.

(d) How many of each type of jeans should be produced per day to maximise profit?

(e) What is the maximum profit?

(f) Explain how the profit would be affected if the profit function was $y=-x+\frac{P}{150}$.

Four points, A, B, C and D on the xy plane are joined with a straight line. (a) Calculate the length of AD.

(b) Find the coordinates of the mid-point of BD.

(c) The point E lies on the x-axis such that BDE is an isosceles triangle.
Angle DBE = angle DEB.
Find the coordinates of point E.

The village chief calls a total of x women and y men to a village gathering.
He expects:

In the village there are:

Two of the inequalities are x + y ≤ 80 and x < 60.
(a) Write the other two inequalities.

(b) By shading the unwanted region, represent the four inequalities on the graph provided.
One of the lines is already drawn for you.

(c) The total number of men and women who finally show up at the gathering is given by the minimum value of the expression 3y + x.
Find the number of men and women who showed up.

The region, R, in the grid represents the number of white (w) and brown (b) loaves of bread produced in a bakery.
One of the inequalities defining the region, R, is $b\le 25-\frac{5}{6}w$. (a) Write the other inequalities defining region R.

(b) If the profit made on a loaf of white bread is M1.80 and the profit made on a loaf of brown bread is M1.40, write an equation for the total profit, P.

(c) Calculate the maximum profit that can be made on selling white and brown loaves of bread.

(d) In one month the profit on a loaf of white bread decreased to M1.60 and on a loaf of brown bread increased to M1.60.
(i) Determine the new profit equation.

(ii) Calculate the profit made by selling 15 loaves of white bread and 10 loaves of brown bread.

(e) State if the new profit equation will affect the maximum profit.
Support your answer by showing all the calculations.