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Problem solving with linear inequalities

I can use my knowledge of inequalities to solve problems.

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Lesson 7 of 7

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Year 11

Problem solving with linear inequalities

I can use my knowledge of inequalities to solve problems.

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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

Keywords

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

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.

Consider having pupils investigate what additional capacity would mean by changing the constraints. What options are available if the constraints still existed but had less restrictive values?

Teacher tip

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.
$$15x+25y=1500$$ is an equation whereas $$15x+25y<1500$$ is an .

Correct Answer: inequality

Q2.
Which of these is the point of intersection of these two linear equations?

$$(0, 0)$$

$$(0, 4)$$

$$(4, 0)$$

Correct answer: $$(4, 12)$$

$$(12, 4)$$

Q3.
For the equation $$c=10+6m$$ what is the value of $$c$$ when $$m=9$$? $$c=$$ .

Correct Answer: 64

Q4.
Solve $$5x<x+28$$.

$$x>7$$

Correct answer: $$x<7$$

$$x<28$$

$$x<5$$

$$x>5$$

Q5.
Which of these values satisfy this inequality? $${3x+6y}\le{450}$$

Correct answer: $$(60,30)$$

Correct answer: $$(30,60)$$

Correct answer: $$(70,20)$$

$$(20,70)$$

$$(60,50)$$

Q6.
A taxi firm charge a fixed rate of $$£4$$ per journey plus $$£1.50$$ per mile travelled. The cost of your journey cannot exceed $$£16$$. Which inequality models this scenario?

$$4m+1.5<16$$

$$4+1.5m<16m$$

$$4+1.5m<16$$

Correct answer: $${4+1.5m}\le16$$

$${4m+1.5}\le16$$

Assessment exit quiz

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6 Questions

Q1.
The graph shows the charges of three taxi firms. The coordinate pair $$(3,9)$$ is the point of __________ of the graphs of taxi firms 'a' and 'b'.

Correct answer: intersection

intercept

inequality

Q2.
In what region does taxi firm 'a' charge less than taxi firm 'b'?

Correct answer: $$m<3$$

$$m>3$$

$$m\le3$$

$$m<4$$

$$m>4$$

Q3.
In what region does taxi firm 'c' charge less than taxi firm 'a'?

$$m<4$$

Correct answer: $$m>4$$

$$m\geq4$$

$$m<8$$

$$m>8$$

Q4.
These inequalities model a company's packaging and production constraints. Which of these points satisfies the production constraint but not the packaging constraint?

$$(10, 80)$$

$$(60, 80)$$

Correct answer: $$(60, 10)$$

$$(10, 10)$$

Q5.
These inequalities model a company's packaging and production constraints. Which of these points satisfy both the production and the packaging constraints?

Correct answer: $$(50,15)$$

$$(15,50)$$

$$(40,30)$$

$$(30,40)$$

Correct answer: $$(60,0)$$

Q6.
These inequalities model a company's packaging and production constraints. Which coordinate pair is the point where both production and packaging are optimised?

Correct Answer: (50,20), (50, 20)

Lesson appears in

UnitMaths / Inequalities

Maths

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Problem solving with linear inequalities

Problem solving with linear inequalities

Lesson slides

Lesson details

Key learning points

Keywords

Common misconception

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.$$15x+25y=1500$$ is an equation whereas $$15x+25y<1500$$ is an .

Q2.Which of these is the point of intersection of these two linear equations?

Q3.For the equation $$c=10+6m$$ what is the value of $$c$$ when $$m=9$$? $$c=$$ .

Q4.Solve $$5x<x+28$$.

Q5.Which of these values satisfy this inequality? $${3x+6y}\le{450}$$

Q6.A taxi firm charge a fixed rate of $$£4$$ per journey plus $$£1.50$$ per mile travelled. The cost of your journey cannot exceed $$£16$$. Which inequality models this scenario?

Assessment exit quiz

6 Questions

Q1.The graph shows the charges of three taxi firms. The coordinate pair $$(3,9)$$ is the point of __________ of the graphs of taxi firms 'a' and 'b'.

Q2.In what region does taxi firm 'a' charge less than taxi firm 'b'?

Q3.In what region does taxi firm 'c' charge less than taxi firm 'a'?

Q4.These inequalities model a company's packaging and production constraints. Which of these points satisfies the production constraint but not the packaging constraint?

Q5.These inequalities model a company's packaging and production constraints. Which of these points satisfy both the production and the packaging constraints?

Q6.These inequalities model a company's packaging and production constraints. Which coordinate pair is the point where both production and packaging are optimised?

Lesson appears in

UnitMaths / Inequalities

Maths

Q1.
$$15x+25y=1500$$ is an equation whereas $$15x+25y<1500$$ is an .

Q2.
Which of these is the point of intersection of these two linear equations?

Q3.
For the equation $$c=10+6m$$ what is the value of $$c$$ when $$m=9$$? $$c=$$ .

Q4.
Solve $$5x<x+28$$.

Q5.
Which of these values satisfy this inequality? $${3x+6y}\le{450}$$

Q6.
A taxi firm charge a fixed rate of $$£4$$ per journey plus $$£1.50$$ per mile travelled. The cost of your journey cannot exceed $$£16$$. Which inequality models this scenario?

Q1.
The graph shows the charges of three taxi firms. The coordinate pair $$(3,9)$$ is the point of __________ of the graphs of taxi firms 'a' and 'b'.

Q2.
In what region does taxi firm 'a' charge less than taxi firm 'b'?

Q3.
In what region does taxi firm 'c' charge less than taxi firm 'a'?

Q4.
These inequalities model a company's packaging and production constraints. Which of these points satisfies the production constraint but not the packaging constraint?

Q5.
These inequalities model a company's packaging and production constraints. Which of these points satisfy both the production and the packaging constraints?

Q6.
These inequalities model a company's packaging and production constraints. Which coordinate pair is the point where both production and packaging are optimised?