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Home > Maths I > Relationships > Simple equations and inequalities

Maths I

Simple equations and inequalities

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Solving inequalities

Introduction

With some inequalities, we deal with one number being less than another number, or with one number being greater than another number.

  • < stands for 'is less than'

  • > stands for 'is greater than'

For example, 4 < 5 and 3 > 2

For the inequality \;x < 3, {x} can be any number less than {3}\;\;.

Example 1

Solve the inequality 2x < 5.

2 times x is less than 5.

Divide both sides by 2.

x is less than \frac{5}{2}.

So the solution is:

x < \frac{5}{2}.

So x is any number less than \frac{5}{2}.

Example 2

Solve the inequality 3x + 1 > 7.

We want the x term on the left-hand side, by itself.

So, take 1 away from the left-hand side.

To balance the equation, we must do the same to the right hand side.

So, take 1 away from the right-hand side.

This gives us, 3x + 1 - 1 > 7 - 1

So, 3x > 6.

x > \frac{6}{3} = 2

The solution is x > 2.

Example 3

Solve the inequality 4a - 2 < 5.

Add 2 to both sides, to leave 4a on the left, by itself:

4a - 2 + 2 < 5 + 2

So, 4a < 7.

Therefore, the solution is a < \frac{7}{4}.

Example 4

Solve the inequality 2(y + 5) < 16.

Either:

2y + 10 < 16 (multiplying out the brackets.)

So, 2y + 10 - 10 < 16 - 10 (subtracting 10 from both sides.)

2y < 6

Therefore, y < 3.

Or:

Divide both sides of 2(y + 5) < 16 by 2, to get

y + 5 < 8.

Therefore y < 3

Now, try to solve these inequalities.

Question

Solve 4a < 9.

Answer

a < \frac{9}{4}

Question

Solve 2y + 6 > 14

Answer

y > 4 (because 2y > 8.)

Question

Solve3x - 5 < 1.

Answer

x < 2 (because 3x < 6.)

Question

Solve 3(4 + x) < 15.

Answer

x < 1 (either from 12 + 3x < 15 and 3x < 3, or from 4 + x < 5)

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