Linear Inequalities and Number Lines

The most critical difference between equations and inequalities is the negative coefficient rule. If you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed (e.g., $<$ becomes $>$, and $\ge$ becomes $\le$). This rule exists because negative numbers run in the opposite direction on the number line; for example, $1 < 2$ is true, but multiplying by $-1$ gives $-1$ and $-2$, and $-1 > -2$.

Crucially, inequalities have additive immunity: you must NEVER reverse the sign when merely adding or subtracting negative numbers. Furthermore, you must NOT multiply or divide an inequality by a variable (like $x$) unless its sign is definitively known, because it could be negative and require a sign flip.

Worked Examples: Calculating One-Variable Inequalities

Example 1: Two-step inequality with the negative rule Calculate the solution for: $5 - 2x \le 21$

• Step 1: Subtract 5 from both sides using inverse operations. Additive immunity applies, so the sign stays the same. $-2x \le 16$

• Step 2: Divide both sides by -2. Apply the negative coefficient rule and reverse the inequality sign. $x \ge -8$

Example 2: Double inequality representing an error interval Calculate the solution for: $4 > 2 - \frac{x}{3} \ge -5$

• Step 1: Subtract 2 from all three parts simultaneously. $2 > -\frac{x}{3} \ge -7$

• Step 2: Multiply all parts by 3 to clear the fraction. $6 > -x \ge -21$

• Step 3: Multiply all parts by -1. You must reverse BOTH inequality signs. $-6 < x \le 21$

If asked for integer solutions (whole numbers), carefully check the endpoints. For $-6 < x \le 21$, $-6$ is not included, so the smallest integer is $-5$. Inequality notation is also used to represent an error interval, which is the range of values a number could have been before being rounded or truncated.

Representing Inequalities on a Number Line

How can you draw an infinite list of answers without taking up the whole page? We sketch solution sets on a number line using specific conventions to show the endpoint of the data range.

A strict inequality ($<$ or $>$) is represented by an open circle (a hollow ring) at the endpoint. This indicates that the boundary value itself is NOT included in the solution set. An inclusive inequality ($\le$ or $\ge$) is represented by a closed circle (a solid, filled-in dot), indicating that the boundary value IS included. For single inequalities, an arrow points left (less than) or right (greater than). For double inequalities, a single horizontal line connects the two circles.

Sketch Example: Representing $-3 \le x < 2$

       ●-------------------------○
  <----|----|----|----|----|----|----|---->
      -4   -3   -2   -1    0    1    2    3

• Number Line: A straight horizontal line marked with integers from $-4$ to $3$.
• Lower Endpoint: A closed circle (●) drawn accurately at $-3$ (inclusive, $\le$).
• Upper Endpoint: An open circle (○) drawn accurately at $2$ (strict, $<$).
• Solution Set: A straight, horizontal connecting line drawn directly between the two circles.

Linear Inequalities in Two Variables (Graphical)

In AQA geometry, you often gain marks for shading in the answers you don't want! When solving inequalities with two variables ($x$ and $y$), the first step is to rearrange the inequality into the form $y = mx + c$. For example, to solve $2x + y \le 6$, you rearrange it to $y \le -2x + 6$. We then graph this as a boundary line on a coordinate grid.

The style of the boundary line depends on the inequality type. Use a solid line for inclusive inequalities ($\le$ or $\ge$) and a dashed or dotted line for strict inequalities ($<$ or $>$). To determine which side of the line forms your solution set, test a coordinate not on the line, such as $(0,0)$. If substituting $(0,0)$ into the inequality makes mathematical sense (e.g., $0 \le 6$), then that side is the "wanted" region.

AQA strictly prefers students to shade the UNWANTED region. By shading out the areas that do not satisfy the inequalities, you leave a clear, unshaded polygon. This overlapping "wanted" area must be explicitly labelled as Region R to secure full marks.