Solving And Graphing Inequalities

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Notes

Representing Inequalities as Regions

A **2D inequality** (e.g. $y < x, x + y \ge 8)$ has a solution region in the xy-plane.
To draw the boundary line, replace the inequality sign with '=' and draw that line.
Use a **solid line** $for \le or \ge (line$ included); use a **dotted line** $for < or > (line$ not included).
For $y \le$ ... or $y <$ ..., the wanted region is **below** the line; for $y \ge$ ... or $y >$ ..., it is **above**.
For vertical lines: $x < k$ → left of $x =$ k; $x > k$ → right of $x = k$ .
If unsure, **test a point** (e.g. (0,0)) to see which side satisfies the inequality.
**Shade the unwanted sides** of each line to leave the wanted region clear; label it **R**.

Finding Inequalities from Regions

Identify the equation of each boundary line $(use y = mx + c$ or $x = k, y = k)$ .
Note whether the line is **solid** $(\le or \ge )$ or **dotted** $(< or >)$ .
If the shaded region is **below** the line, $use \le or$ <; if **above**, $use \ge or >$ .
For vertical lines: region **left** of $x = k$ → $\le or$ <; **right** → $\ge or >$ .
**Test a point** from the shaded region to confirm the inequality sign.
Write all inequalities together as the final answer.

Solving Linear Inequalities

Solve inequalities similarly to equations, but **reverse the inequality sign** when multiplying/dividing by a negative number.
Example: $3x + 12 < 5x - 3$ → subtract $3x: 12 < 2x - 3$ → add $3: 15 < 2x$ → $x > 7.5$ .
Example: $3n - 5 > 17 + 8n$ → subtract $3n: -5 > 17 + 5n$ → subtract $17: -22 > 5n$ → $n < -4.4$ .
Always check your solution by substituting a value back into the original inequality.

Shading Unwanted Regions (Exam Technique)

Read carefully: the question may ask you to shade **unwanted** regions or the **wanted** region.
Shading unwanted regions leaves the **wanted region unshaded** (clear).
Draw all boundary lines first, then shade each unwanted side.
Label the final wanted region with the letter **R**.
Use a **test point** (e.g. (0,0)) to decide which side to shade for each inequality.

Integer Solutions in a Region

Sometimes you need to find integer coordinates (x, y) inside a region that satisfy an additional equation.
List all integer points inside the region and check which satisfy the given equation.
Example: find (x, y) with integer coordinates inside R such that $3x + 5y = 35$ .

Word Problems: Forming Inequalities

Translate conditions into inequalities: 'fewer than 10 mats' → $y <$ 10; 'at least 15 silver balloons' → $x \ge 15$ .
'More gold than silver' → $y >$ x; 'total no more than 70' → $x + y \le 70$ .
For time constraints: e.g. 2.25 hours per basket and 1.5 hours per mat, max 22.5 hours → $2.25x + 1.5y \le 22.5$ → multiply by $4: 9x + 6y \le 90$ → divide by $3: 3x + 2y \le 30$ .
Always define variables clearly (e.g. $x =$ number of baskets, $y =$ number of mats).

Graphing Multiple Inequalities

Draw all boundary lines on the same axes using correct line styles (solid/dotted).
Shade the **unwanted** region for each inequality; the remaining unshaded area is the solution region.
Label the solution region **R**.
Check that a point inside R satisfies all inequalities.

Shading Unwanted Regions for Three Inequalities

Finding Inequalities from a Shaded Region

Solving a Linear Inequality Graphically

Word Problem: Feasible Region

Practice questions

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1.Which inequality represents the region below the line $y = 3x + 2$ , including the line?
Easy
- A $y \le 3x + 2$
- B $y < 3x + 2$
- C $y \ge 3x + 2$
- D $y > 3x + 2$
2.Which line would be drawn as a dotted line when graphing the inequality $y > 2x - 1$ ?
Easy
- A $y = 2x - 1$
- B $y = 2x + 1$
- C $y = -2x - 1$
- D $y = -2x + 1$
3.The inequality $x \ge 3$ is represented on a graph by a vertical line at $x = 3$ . Which side is the wanted region?
Easy
- ATo the right of the line
- BTo the left of the line
- CAbove the line
- DBelow the line
4.Solve the inequality $3x + 12 < 5x - 3$ .
Medium
- A $x > 7.5$
- B $x < 7.5$
- C $x > 3$
- D $x < 3$
5.Solve the inequality $3n - 5 > 17 + 8n$ .
Medium
- A $n < -4.4$
- B $n > -4.4$
- C $n < 4.4$
- D $n > 4.4$
6.Solve the inequality $\frac{x}{2} - 13 > 12 + 3x$ .
Medium
- A $x < -10$
- B $x > -10$
- C $x < 10$
- D $x > 10$
7.Which inequality is represented by the solid line $y = 2x + 3$ and the region above it?
Medium
- A $y \ge 2x + 3$
- B $y > 2x + 3$
- C $y \le 2x + 3$
- D $y < 2x + 3$
8.A region is defined by $y \le 2, x \ge 1$ , and $y \ge x$ . Which point lies inside the region?
Medium
- A(2, 1)
- B(0, 0)
- C(3, 4)
- D(1, 3)

Summary

## Solving & Graphing Inequalities (CIE IGCSE Extended) ### Key Rules - **Solid line** for ≤ or ≥ (line included). - **Dotted line** for < or > (line not included). - For y ≤ ... or y < ... : wanted region **below** the line. - For y ≥ ... or y > ... : wanted region **above** the line. - For x < k : wanted region **left** of vertical line x = k. - For x > k : wanted region **right** of vertical line x = k. - **Shade unwanted regions** to leave the wanted region clear; label it **R**. ### Solving Linear Inequalities Treat like an equation, but **flip the inequality sign** when multiplying/dividing by a negative number. ### Finding Inequalities from a Graph 1. Find the equation of each boundary line (use y = mx + c or x = k). 2. Check line type: solid (≤ or ≥) or dotted (< or >). 3. Test a point in the shaded region to decide direction. ### Common Question Types - **Shading regions** defined by multiple inequalities. - **Writing inequalities** from a shaded region. - **Word problems** (e.g., constraints on baskets/mats, cars, balloons) – translate conditions into inequalities, then graph. ### Exam Tips - Always read whether the diagram shades the **wanted** or **unwanted** region. - Use a test point (e.g., (0,0)) if unsure which side to shade. - For integer solutions, list all integer pairs satisfying all inequalities.

Key terms

Inequality: A mathematical statement comparing two expressions using <, >, ≤, or ≥.
Region: The set of points (x,y) that satisfy one or more inequalities, often shown as a shaded area on a graph.
Solid line: Used on a graph to indicate that the line itself is included in the region (for ≤ or ≥).
Dotted line: Used on a graph to indicate that the line itself is not included in the region (for < or >).
Unwanted region: The part of the graph that does NOT satisfy the inequality; it is shaded to leave the wanted region clear.

Slides

What are 2D Inequalities?

A 2D inequality involves two variables, e.g. $y < x$ or $x + y \ge 8$ .
The solution is a region in the xy-plane.
A 1D inequality like $y \ge 2$ also gives a region (points on or above $y = 2)$ .

Which of the following is a 2D inequality?

$x > 5$
$y < x + 2$
$y = 3$
$x + 2 = 7$

Drawing Inequalities on a Graph

Replace the inequality sign with '=' and draw that line.
Use a solid line $for \le or \ge (line included)$ .
Use a dotted line $for < or > (line$ not included).
For $y \le$ ... or $y <$ ... shade below the line; for $y \ge$ ... or $y >$ ... shade above.

For the inequality $y < 2x$ , the line should be drawn as $a ____$ line.

Vertical and Horizontal Lines

For vertical lines: $x < k$ → region to the left; $x > k$ → region to the right.
For horizontal lines: $y < k$ → region below; $y > k$ → region above.
Use a solid line $if \le or \ge$ , dotted $if < or >$ .

Which of these are true about shading?

For $y \le 3$ , shade below the solid line $y = 3$ .
For $x > 2$ , shade to the left of the dotted line $x = 2$ .
For $y \ge x + 1$ , shade above the solid line.

Shading the Unwanted Region

Often you shade the unwanted sides of each line.
This leaves a clear region that satisfies all inequalities.
Label the wanted region with the letter R.

When shading unwanted regions, what does the unshaded area represent?

The region that satisfies none of the inequalities
The region that satisfies all inequalities
The region that satisfies only one inequality
The region outside the graph

Worked Example: Representing Inequalities

Consider: $3x + 2y \ge 12, y < 2x, x < 3$ .
Draw solid line for $3x + 2y = 12 (rearrange$ to $y = -1.5x + 6)$ .
Draw dotted lines for $y = 2x$ and $x = 3$ .
Shade unwanted regions: below $3x+2y=12$ , above $y=2x$ , right of $x=3$ .
Label the remaining region R.

For the inequality $y < 2x$ , the point (1,0) lies in $the ____$ region.

Finding Inequalities from a Graph

Identify each line (find equation using gradient and intercept).
Note if line is solid $(\le or \ge )$ or dotted $(< or >)$ .
Determine inequality sign: above line → $\ge or$ >; below → $\le or <$ .
For vertical lines: left → $\le or$ <; right → $\ge or >$ .

A graph shows a solid line $y = 2x + 1$ and the shaded region is below the line. Which inequality is represented?

$y < 2x + 1$
$y \le 2x + 1$
$y > 2x + 1$
$y \ge 2x + 1$

Worked Example: Finding Inequalities

Given a shaded region, find the three inequalities.
Lines: $x = 1 (solid), y = x (dotted), y = -x + 7 (solid)$ .
Shaded region: above $y = x$ , below $y = -x + 7$ , right of $x = 1$ .
Inequalities: $y > x, y \le -x + 7, x \ge 1$ .

Which statements correctly describe the inequalities from the worked example?

$y > x$ because the line is dotted and region is above.
$y \le -x + 7$ because the line is solid and region is below.
$x \ge 1$ because the line is solid and region is to the left.

Checking with a Point

If unsure about which side to shade, test a point (e.g., (0,0)).
Substitute into the inequality; if true, that point is in the wanted region.
If false, shade the other side.

To check the inequality $y \ge 2x - 3$ , you test point (0,0). $0 \ge -3$ is true. What does this tell you?

(0,0) is in the unwanted region
(0,0) is in the wanted region
The line should be dotted
The inequality is wrong

Summary: Solving & Graphing Inequalities

Draw lines (solid $for \le /\ge$ , dotted $for </>)$ .
Shade unwanted regions (or wanted if specified).
Label the region R.
To find inequalities from a graph, find line equations and test a point.

A dotted line indicates the boundary $is ____$ included in the region.

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