Advanced problem solving with linear inequalities | Maths

Starter quiz

$5x+15y=1200$ is an equation whereas $5x+15y<1200$ is an ______.

'inequality' ✓
$(5,2)$ is the point of ______ of these two linear equations.

'intersection' ✓
A young person starts with $\pounds50$ savings and then saves $\pounds15$ per week. Which inequality represents the time when they will have more than $\pounds500$ in savings?

$50w+15\geq500$

$50w+15>500$

$50w+15<500$

$50+15w\geq500$

$50+15w>500$ ✓
Which of these values satisfy this inequality? ${8x+12y}\le{480}$

$(15,30)$ ✓

$(30,15)$ ✓

$(40,10)$ ✓

$(10,40)$

$(40,15)$
Which of these values satisfy both of these inequalities simultaneously? $y>2x+3$ and $y<12$

$(2,8)$ ✓

$(8,2)$

$(3,10)$ ✓

$(3,9)$

$(2,12)$
Which of these values satisfy both of these inequalities simultaneously? $y>2x+3$ and $y\le12-x$

$(0,12)$ ✓

$(10,2)$

$(2,10)$ ✓

$(3,9)$

$(1,6)$ ✓

Exit quiz

These inequalities model a company's packaging and production constraints. The values $(15,20)$ satisfy both inequalities __________.

simultaneously ✓

independently

individually
These inequalities model a company's packaging and production constraints. Which of these points satisfies the production constraint but not the packaging constraint?

$(5,40)$

$(40,40)$

$(40,5)$ ✓

$(5,5)$
These inequalities model a company's packaging and production constraints. Which of these points satisfy both the production and the packaging constraints?

$(5,35)$

$(35,5)$ ✓

$(10,30)$

$(30,10)$ ✓

$(15,30)$
These inequalities model a company's packaging and production constraints. Which coordinate pair is the point where both production and packaging are optimised? ______

'(30,15)' ✓
This model has three constraints. $x$ represents the number of standard games produced, $y$ the number of deluxe ones. What is the maximum number of games that can be made? ______

'18' ✓
$This model has three constraints. <Math>x =</Math> standard games, making <Math>\pounds4</Math> profit. <Math>y =</Math> deluxe games, making <Math>\pounds9</Math> profit. What is the maximum profit achievable? <Math>\pounds</Math><span class="blank">______</span>$

This model has three constraints. $x =$ standard games, making $\pounds4$ profit. $y =$ deluxe games, making $\pounds9$ profit. What is the maximum profit achievable? $\pounds$ ______

'104' ✓

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Lesson Details

Key learning points

When handling a project, there are often constraints.
Constraints exist for a variety of reasons, such as cost or supply limitations.
Finding a set of possible solutions allows you to work within the constraints.

Common misconception

We are only interested in points that are in the region satisfied by all inequalities.

With practical contexts, such as project management, it is useful to know when one constraint is not being met as it can suggest where additional resources (if available) should be allocated.

Keywords

Inequality - An inequality is used to show that one expression may not be equal to another.