GCSE · Maths · Edexcel · Spec 1MA1 · Foundation+Higher

Solving linear and quadratic inequalities

An equation gives one answer. An inequality gives a whole range, and one sneaky rule can flip it the wrong way round.

Maths · Inequalities

Mark every value x could be

Solve 5 ≤ x + 3 < 24 by doing the same to every part. Then mark every value x could be: set both ends, and decide whether each end is included.

Exam line: Give the answer as a range, then check each end: ≤ and ≥ ends are included (filled circle), < and > ends are not (open circle).

Predict, then check

No algebra yet: just the number line.

Start with 2 < 3, which is true. Now multiply both sides by −1. Which sign goes between −2 and −3?

Maths · Linear inequalities

Supply the missing steps

Solve 5 − 9z ≤ 22, collecting z so that its coefficient stays positive.

  1. 5 − 9z ≤ 22The z term is negative, so move it to the other side.
  2. missing step
Which line is step 2?

Solve x² − 2x < 3: where is the curve below zero?

-2-0.512.54-6-3036xy = x² − 2x − 3(1, -4)

x: 1. y = x² − 2x − 3: -4

slide the point along the curve and watch y

Exam line: Make one side zero: x² − 2x − 3 < 0. Factorise: (x + 1)(x − 3), so the roots are x = −1 and x = 3. The curve is below zero between the roots, so x² − 2x − 3 < 0 gives −1 < x < 3. It is above zero outside them, so x² − 2x − 3 > 0 gives x < −1 or x > 3.
Watch out: The sign decides between one set and two. Less than zero gives one set between the roots (a closed set if it is ≤). More than zero gives two separate sets, one below the smaller root and one above the larger.

Maths · Regions

Is the point in the region?

Three inequalities must all hold: y ≥ 2 (solid line), y > x (dashed line) and x + y ≤ 8 (solid line). Substitute each point into all three, then sort it.

Still to sort

In the solution region (0)

Satisfies all three inequalities.

Where the line is: A point on a solid line counts, because ≤ and ≥ include it.

Not in the region (0)

Fails at least one inequality.

Where the line is: A point on a dashed line does not count, and neither does one that fails just a single inequality.

6 of 6 still to sort.

Exam line: Substitute each point into every inequality. A solid line includes its points; a dashed line does not.
Watch out: The solution region is not automatically the area inside the shape the lines form. One failed inequality is enough to rule a point out.

Maths · Constraints

Write it, then mark it

Reminder: write each limit as an inequality, make the units match first, and graph the boundary lines. Where two boundary lines cross, the point meets both limits exactly.

A company makes x standard games and y deluxe games. A standard game takes 20 minutes to make and a deluxe game takes 30 minutes. The machine can run for at most 20 hours, and the company must make at least 10 deluxe games. Write down an inequality for each limit, then say what it means where the two boundary lines cross. [4 marks]

0 words · your answer stays on this page and is not sent anywhere.

Exam line: Match the units first (20 hours is 1200 minutes), write one inequality per limit, and remember the crossing point of two boundary lines is where both limits are reached.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

An answer that is a whole range of numbers

What you need to know

  • An inequality uses <, >, ≤ or ≥ instead of =, and its solution is usually a range of values, not one number.
  • Solve a linear inequality like an equation: do the same to both sides until the variable is on its own.
  • Have a goYour friend says 7 + 3k > 25 “can’t be solved because it’s got a > in it”. Prove them wrong: what does k have to be?

    k > 6

    Subtract 7 from both sides to get 3k > 18, then divide both sides by 3. Writing k = 6 is the tempting slip: the answer is a range, and k = 6 itself fails because 25 is not greater than 25.

  • Multiplying or dividing both sides by a negative number reverses the sign, so 2 < 3 becomes −2 > −3. Adding the negative term to both sides avoids it.
  • Have a goA classmate solves −4x > 12, writes x > −3, and says “dividing never changes the sign”. Confident, but wrong. What should the answer be?

    x < −3

    Dividing both sides by −4, a negative number, reverses the sign. Check x = −4: −4 × −4 = 16, which is greater than 12, but x > −3 would wrongly leave −4 out.

  • A double inequality like 2 ≤ x < 21 is two inequalities in one: solve each part, or do the same to all parts.
  • On a number line, a filled circle means the end is included (≤ or ≥); an open circle means it is not (< or >).
  • Have a goIs x = −1 a solution of −1 ≤ x ≤ 3? And what about x = 3.5?

    Yes to −1. No to 3.5.

    Both ends have ≤, so both ends are included and get filled circles. 3.5 lies beyond the upper end at 3, so it is outside the range.

  • For a quadratic inequality, make one side zero, find the roots (by factorising, say), then sketch the graph and read off the solution.
  • With a positive x² term, the curve is below zero between the roots and above zero outside them.
  • In two variables, a boundary line is solid for ≤ or ≥ and dashed for < or >.
  • Test a point to find each side. The solution region satisfies every inequality, so it is not automatically the inside of a shape.
  • For constraint problems, write each limit as an inequality in matching units. Where two boundary lines cross, both limits are reached.

The big picture

Inequalities use <, >, ≤ and ≥, and their solutions are usually a range of values. Solve linear ones like equations, but reverse the sign when you multiply or divide both sides by a negative number. Quadratic inequalities are read off a sketch of the curve, and inequalities in two variables become regions, checked by testing points.

Key points

1A solution to an inequality is usually a range of values, written with <, >, ≤ or ≥ or shown on a number line.
2Solve linear inequalities with the same moves as an equation, but reverse the sign when you multiply or divide both sides by a negative number.
3For a double inequality, do the same to every part. Some combined inequalities have no valid solution.
4For a quadratic inequality, make one side zero, find the roots, sketch, and read the sets off the curve. With a positive x² term, it is below zero between the roots and above zero outside them.
5In two variables, boundary lines are solid for ≤ or ≥ and dashed for < or >. Test a point for each side, because the region must satisfy every inequality.
6If x and y are integers, you can list the coordinate pairs that lie in the region by substituting them.
7In constraint problems, write each limit as an inequality in matching units. The point where two boundary lines cross is where both limits are reached.

Worked example

Problem

Solve 80 + 5w > 10w.

⚠ Watch out

Treating an inequality exactly like an equation. Solving −2x < 10 by dividing by −2 and writing x < −5 forgets the reversal: dividing by a negative number flips the sign, so the answer is x > −5.

🧠

Memory hook

Times or divide by a negative? Flip the sign. And solid line means included, dashed line means not.

✓

Check yourself

Solve x² − 5x + 4 > 0. One set or two? Roots 1 and 4, curve above zero outside them: x < 1 or x > 4.

Flashcards

(13)
What do the four inequality symbols mean?
< less than, > greater than, ≤ less than or equal to, ≥ greater than or equal to. The answer to an inequality is usually a range of values.
How do you solve a linear inequality?
Like an equation: do the same to both sides (add or subtract, then multiply or divide) until the variable is on its own.
What happens to the sign when you multiply or divide both sides by a negative number?
It reverses. 2 < 3 becomes −2 > −3 when both sides are multiplied by −1.
Why does the sign reverse when you multiply by −1?
Multiplying by −1 reflects the number line at 0, so the order of the numbers flips.
How can you avoid reversing the sign?
Add the term with the negative variable to both sides, so the variable ends up with a positive coefficient.
Solve 6 + 2m < 4m.
Subtract 2m from both sides: 6 < 2m. Divide by 2: 3 < m, so m > 3.
What does 2 ≤ x < 21 mean, and how do you solve a double inequality?
x is at least 2 and less than 21. Solve it as two separate inequalities, or do the same operation to all the parts.
On a number line, which ends get a filled circle?
Ends with ≤ or ≥ are included, so they are filled. Ends with < or > are not included, so they are open.
What are the steps for a quadratic inequality?
Make one side zero, find the roots (by factorising, say), sketch the graph, then read off the solution set.
A quadratic has a positive x² term. Where is it below zero, and where above?
Below zero between the roots (one set). Above zero outside the roots (two sets, one each side).
When is a boundary line solid, and when dashed?
Solid for ≤ or ≥. Dashed for < or >, because points on a dashed line do not satisfy the inequality.
How do you decide which side of a line is the solution?
Test a point: substitute it and see if the inequality is true. The solution region satisfies all the inequalities together.
In a constraints problem, what does the point where two boundary lines cross tell you?
Both constraints are exactly at their limit at that point.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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