Solving Linear Inequalities (Edexcel IGCSE Maths)

Spec point — 2.8D

Solving linear inequalities

What is an inequality?

• An inequality tells you that something is greater than (>) or less than (<) something else

  • x > 5 means x is greater than 5

    • x could be 6, 7, 8, 9, ...

• Inequalities may also include being equal (=)

  • ⩾ means greater than or equal to

  • ⩽ means less than or equal to

    • x ⩽ 10 means x is less than or equal to 10

      • x could be 10, 9, 8, 7, 6,....

• When they cannot be equal, they are called strict inequalities

  • and < are strict inequalities

    • x > 5 does not include 5 (strict)
    • x ⩾ 5 does include 5 (not strict)

How do I find integers that satisfy inequalities?

• You may be given two end points and have to list the integer values of x that satisfy the inequality

  • Look at whether each end point is included or not

    • 3 ⩽ x ⩽ 6

      • x = 3, 4, 5, 6

    • 3 ⩽ x < 6

      • x = 3, 4, 5

    • 3 < x ⩽ 6

      • x = 4, 5, 6

    • 3 < x < 6

      • x = 4, 5

• If only one end point is given, there are an infinite number of integers

  • x > 2

    • x = 3, 4, 5, 6, ...

  • x ⩽ 2

    • x = 2, 1, 0, -1, -2, ...
    • Remember zero and negative whole numbers are integers
    • If the question had said positive integers only then just list x = 2, 1

• You may be asked to find integers that satisfy two inequalities

  • 0 < x < 5 and x ⩾ 3

    • List separately: x = 1, 2, 3, 4 and x = 3, 4, 5, 6, ...
    • Find the values that appear in **both **lists: x = 3, 4

• If the question does not say x is an integer, do not assume x* is an**** integer!

  • x > 3 actually means any value greater than 3

    • 3.1 is possible
    • $π$ = 3.14159... is possible

• You may be asked to find the smallest or largest integer

  • The smallest integer that satisfies x > 6.5 is 7

Worked Example List all the integer values of $x$ that satisfy

$-4\leq x<2$

Answer:

Integer values are whole numbers -4 ≤ x shows that x includes -4, so this is the first integer

x = -4*

x < 2 shows that x does not include 2 Therefore the last integer is x = 1

x = 1*

For the answer, list all the integers from -4 to 1 Remember integers can be zero and negative

$x=-4,-3,-2,-1,0,1$

How do I represent an inequality on a number line?

• The inequality -3 < x ≤ 4 is shown on a number line below

• Draw circles above the end points and connect them with a horizontal line

  • Leave an open circle for end points with strict inequalities, < or >

    • These end points are not included

  • Fill in a solid circle for end points with ≤ or ≥ inequalities

    • These end points are included

• Use a horizontal arrow for inequalities with one end point

  • x > 5 is an open circle at 5 with a horizontal arrow pointing to the right

Worked Example Represent the following inequalities on a number line.

(a) $-2\leq x<1$

Answer:

-2 is included so use a closed circle

1 is not included so use an open circle

(b) $t<3$

Answer:

3 is not included so use an open circle

There is no second end point Any value less than three is accepted, so draw a horizontal arrow to the left

How do I solve linear inequalities?

• 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 applying the same operation to all three sides of the inequality

  • E.g. $\frac{a}{2}<x<\frac{b}{2}$

• Alternatively, you can separate the inequality into two inequalities

  • $a<2x$ and $2x<b$

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

Final answer: $x\leq13$

Worked Example Solve the inequality $10-2x>4x-7$.

Answer:

Add $2x$ to both sides

$10>6x-7$

Add 7 to both sides

$17>6x$

Divide both sides by 6

$\frac{17}{6}>x\mathrm{or}x<\frac{17}{6}$