Linear Inequalities (AQA GCSE Further Maths): Revision Note

Linear inequalities

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 a y); but no x² terms or terms with higher powers of x

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

• For example, 3x + 4 ⩾ 7 would be read “3x + 4 is greater than or equal to 7”. 

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. 1 < 2 → [times both sides by (–1)] → –1 > –2 (sign flips)

• Never multiply or divide by a variable (x) as this 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 Number Lines, Set Notation and deal with “Double” Inequalities

How do I represent linear inequalities on a number line?

• Inequalities such as $x<a$ and $x>a$ can be represented on a normal number line using an open circle and an arrow

  • For $<$, the arrow points to the left of $a$

  • For $>$, the arrow points to the right of $a$

• Inequalities such as $x\le a$ and $x\ge a$ can be represented on a normal number line using a solid circle and an arrow

  • For $\le$, the arrow points to the left of $a$

  • For $\ge$, the arrow points to the right of $a$

• Inequalities such as $a<x<b$ and $a\le x\le b$ can be represented on a normal number line using two circles at $a$ and $b$ and a line between them

  • For $<$ or $>$ use an open circle

  • For $\le$ or $\ge$, use a solid circle

• Disjoint inequalities such as "$x<a$ or $x>b$" can be represented with two circles at $a$ and $b$, an arrowed line pointing left from $a$ and an arrowed line pointing right from $b$, and a blank space between $a$ and $b$

How do I represent linear inequalities using set notation?

• We use curly brackets and a colon in set notation. $\left\{x:...\right\}$ means "x is in the set ..."

  • For example; if x is greater than 3, then in set notation, $\left\{x:x>3\right\}$

• However, if x is between two values, then the two end values must be written in separate sets, using the intersection symbol, $\cap$

  • For example, if x is greater than 3 and less than or equal to 5, then in set notation, $\left\{x:x>3\right\}\cap \left\{x:x\le 5\right\}$

• Similarly, if x is less than one value or greater than another (disjoint), then the two end values must be written in separate sets using the union symbol, $\cup$

  • For example, if x is less than 3 or greater than or equal to 5, then in set notation, $\left\{x:x<3\right\}\cup \left\{x:x\ge 5\right\}$

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