Cambridge International AS and A Level Mathematics Pure Mathematics 1

Inequalities

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1

Linear inequalities

!   The methods for linear inequalities are much the same as those for equations but

you must be careful when multiplying or dividing through an inequality by a

negative number.

Take for example the following statement:

        5      3    is true

Multiply both sides by −1   –5  −3 is false.

!   It is actually the case that multiplying or dividing by a negative number reverses

the inequality, but you may prefer to avoid the difficulty, as shown in the

examples below.

EXAMPLE 1.37 Solve 5 x − 3  2 x − 15.

SOLUTION

Add     3    to, and subtract   2 x from, both sides    ⇒   5 x −   2 x  − 15   +  3

Tidy up  ⇒      3 x  − 12

Divide both sides by    3    ⇒ x  − 4

Note

Since there was no need to multiply or divide both sides by a negative number, no

problems arose in this example.

EXAMPLE 1.38 Solve 2 y + 6  7 y + 11.

SOLUTION

Subtract    6    and    7 y from both sides  ⇒  2 y −   7 y    11   −  6

Tidy up  ⇒ − 5 y > + 5

Add     5 y to both sides and subtract  5        ⇒ − 5    + 5 y

Divide both sides by + 5     ⇒ − 1    y

Note that logically − 1   y is the same as y  −1, so the solution is y  −1.

Beware: do not

divide both sides

by –5.

This now allows

you to divide both

sides by +5.