Solving Inequalities (Edexcel IGCSE Further Maths)

Solving Linear Inequalities

Spec point — 3D

What is a linear inequality?

• An inequality tells you that one expression is greater than (“>”) or less than (“<”) another

  • “⩾” means “greater than or equal to”
  • “⩽” means “less than or equal to”

• A linear inequality only has constant terms (numbers with no letters) and terms in x (and/or y)

  • but no* x² terms or terms with other powers of x

    • 3x² > 12 is not a linear inequality (it is a quadratic inequality)

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

    • Changing the inequality sign to an equals sign changes 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. 1 < 2

    • Multiply both sides by –1 (negative number)
    • It becomes –1 > –2(sign flips)

• Never multiply or divide by a variable(x)

  • It could be positive or negative

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

• You also need to know how to

  • use set notation
  • deal with “double” inequalities

How do I represent linear inequalities using set notation?

• We use curly brackets and a colon in set notation

  • $\{ x : ... \}$ means "x is in the set such that ..."

    • e.g. the set of all x such that x is greater than 3 is written $\{ x : x > 3 \}$

• If x is between two values

  • you can write it as a single set

    • e.g. if x is greater than 3 and less than or equal to 5, then in set notation $\{ x : 3 < x \le 5 \}$

  • or you can write it as separate sets, using the intersection symbol, $\cap$

    • so the above example could also be written in set notation as $\{ x : x > 3 \} \cap \{ x : x \le 5 \}$

  • Either way will get the marks, unless a question specifically asks for one or the other

• If x is less than one value OR greater than another value (disjoint sets)

  • then the two end values must be written in separate sets using the union symbol, $\cup$

    • e.g. if x is less than 3 or greater than or equal to 5, then in set notation $\{ x : x < 3 \} \cup \{ x : \ge 5 \}$

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

  • You will lose marks in an exam for doing this.

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

Worked Example (a) Solve the inequality $-7\leq3x-1<2$.

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.

$-2\leq x<1$

(b) Write your answer to part (a) in set notation as the intersection of two sets.

Rewrite your answer using the set notation rules discussed above

$\{ x : x \ge - 2 \} \cap \{ x : x < 1 \}$

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

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

$x\geq-8$

The final answer is normally written with the number first, but you won't be penalised for writing the x first so long as the inequality sign is the correct way around

$x\geq-8$ or $-8\leq x$

Solving Quadratic Inequalities

Spec point — 3D

What are quadratic inequalities?

• A quadratic inequality is an inequality with a term in x² (but no higher powers of x)
• Solving the inequality requires solving the corresponding quadratic equation
• Sketching a quadratic graph is essential

How do I solve quadratic inequalities?

• STEP 1 Rearrange the inequality into quadratic form with a positive squared term

  • ax²+ bx + c > 0 with a > 0

    • The inequality sign may be >, <, ≤ or ≥

• STEP 2 Find the roots of the quadratic equation

  • Solve ax² + bx + c = 0 to get x₁* and x₂ where x₁ < x₂

• STEP 3 Sketch a graph of the quadratic and label the roots

  • As the squared term is positive it will be "U" shaped

• STEP 4 Identify the region that satisfies the inequality

  • For ax²+ bx + c > 0 you want the region above the x-axis

    • The solution is x< x₁ or x > x₂
    • It is x≤ x₁ or **x ≥ x₂ **if the original inequality was ≥ instead of >

  • For ax²** + bx + c < 0** you want the region below the x-axis

    • The solution is x > x₁ and **x < x₂ **
    • This is more commonly written as x₁** < x < x₂
    • It is x ≥ x₁ and x ≤ x₂** (x₁ ≤ x ≤ x₂) if the original inequality was ≤ instead of <

• Avoid multiplying or dividing by a negative number

  • If unavoidable, remember to “flip” the inequality sign (so < → >, ≥ → ≤, etc)

• Avoid multiplying or dividing by a variable (x)

  • The variable could be negative

  • Multiplying or dividing by x² is allowed

    • However this can create extra invalid solutions

• Do rearrange to make the x² term positive

Exam Hint

• Always start by rearranging to a quadratic with positive squared term
• Always sketch a graph of the quadratic before deciding the final answer

Worked Example