Rose Harrison, Clara Huizink, Aidan Sproat Clements and, Marlene Torres Skoumal Solutions for Exercise 18: Mixed practice

Solve the following inequality algebraically:

$x<3x-1<2x+7$

Solve the following inequality graphically:

$4-2x\le 2$

Solve the following inequality graphically:

$\frac{x}{3}+5<11$

Solve the following inequality graphically:

$2\left(x+1\right)>x-7$

Solve the following inequality graphically:

$3\left(x+1\right)<2\left(1-x\right)$

A company hires out buses:

A school can spend up to $\mathrm{UK}\pounds 400$ to hire buses to take $120$ students on an outing. Determine how many of each bus the school should hire to minimize the total cost of bus hire.

A factory manager has to decide which machines to install to manufacture a product. 

Machine $A$

Output is $300$ units per week. Needs $500 {\mathrm{m}}^2$ of space, and $10$ operators.

Machine $B$

Output is $200$ units per week. Needs $600 {\mathrm{m}}^2$ of space, and $6$ operators.

The factory has $5050 {\mathrm{m}}^2$ of space available and $80$ operators. Its production target is $2400$ units per week. Determine if the factory can meet its production target. If it can, determine how many of each type of machine it should buy.

Your company has a maximum of $\mathrm{US}\$2200$ to spend on advertising. You plan to run at most $20$ advertisements because each one needs to be different and creative. An advertisement in the weekday newspaper costs $\mathrm{US}\$50$, and in the weekend newspaper costs $\mathrm{US}\$200$. Each weekday advert results in an average of $100$ new customers, and each weekend advert results in an average of $300$ new customers. You want to maximize the number of new customers.

Find the maximum number of weekday and weekend newspaper advertisements you can buy.