Solving Linear Inequalities (Edexcel IGCSE Maths)

Spec point — 2.8D

Solving linear lnequalities

How do I solve a linear inequality?

• Solving linear inequalities is just like Solving Linear Equations

  • Follow the same rules, but keep the inequality sign throughout
  • If you change the inequality sign to an equals sign you are changing the meaning of the problem

• When you multiply or divide both sides by a negative number, you must flip the sign of the inequality

  • E.g. space 1 less than 2 $( \times - 1 ) ( \times - 1 )$ space minus 1 greater than negative 2

• Never multiply or divide by a variable(x) as this could be positive or negative

• The safest way to rearrange is simply to add and subtract to move all the terms onto one side

How do I solve double inequalities?

• Inequalities such as $a<2x<b$ can be solved by doing the same thing to all three parts of the inequality

  • Use the same rules as solving linear inequalities

Exam Hint

• Do not change the inequality sign to an equals when solving linear inequalities.

  • In an exam you will lose marks for doing this.

• Remember to reverse the direction of the inequality sign when multiplying or dividing by a negative number!

Worked Example Solve the inequality $2x-5\leq21$.

Answer:

Add 5 to both sides

$2x\leq26$

Now divide both sides by 2

$x\leq13$

Final answer: $x\leq13$

Worked Example Solve the inequality $5-2x\leq21$.

Answer:

Subtract 5 from both sides, keeping the inequality sign the same

$-2x\leq16$

Now divide both sides by -2. However because you are dividing by a negative number, you must flip the inequality sign

Final answer: $x\geq-8$or$-8\leq x$

Worked Example Solve the inequality $-7\leq3x-1<2$, illustrating your answer on a number line.

Answer:

This is a double inequality, so any operation carried out to one side must be done to all three parts Use the expression in the middle to choose the inverse operations needed to isolate x

Add 1 to all three parts Remember not to change the inequality signs

$-6\leq3x<3$

Divide all three parts by 3 3 is positive so there is no need to flip the signs

$-2\leq x<1$

Illustrate the final answer on a number line, using an open circle at 1 and a closed circle at -2.