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Year 10AC9M10A02

Linear Inequalities on a Graph

Solve linear inequalities and simultaneous linear equations in 2 variables; interpret solutions graphically and communicate solutions in terms of the situation.

An equation has a few answers. An inequality has a whole region of them. The line you draw is only the edge — the answer is everything on one side of it.

Builds onYr 8 · Drawing a Line — Gradient Intercept Method

Method

Solid or dashed
≤, ≥ ⇒ solid<, > ⇒ dashed\leq,\ \geq \ \Rightarrow\ \text{solid} \qquad <,\ > \ \Rightarrow\ \text{dashed}
A solid line means the edge is included. A dashed line means it is not.
Flip when you divide by a negative
−2x>6 ⇒ x<−3-2x > 6 \ \Rightarrow\ x < -3
The single most common error in the whole topic. Multiplying or dividing by a negative reverses the sign.
Test a point to find the side
Use $(0;0)$ whenever the line does not pass through it. If the statement is true, shade that side.
Shade the region that works
Say clearly which convention you are using. Shading the wanted region is the usual one.

Worked example

e.g. Sketch the region y≤2x+1\text{Sketch the region } y \leq 2x + 1
  1. Replace the inequality sign with an equals sign. That gives the boundary line.y=2x+1y = 2x + 1
  2. Draw it using the gradient and the yy-intercept.m=2,c=1m = 2,\quad c = 1
  3. Decide solid or dashed from the original sign.≤ ⇒ solid line\leq \ \Rightarrow\ \text{solid line}
  4. Test the point (0;0)(0;0) in the original inequality.0≤2(0)+1 ⇒ 0≤1true0 \leq 2(0) + 1 \ \Rightarrow\ 0 \leq 1 \quad \text{true}
  5. True means (0;0)(0;0) is inside the region, so shade that side.shade below the line\text{shade below the line}
    The line y = 2x + 1 with the region below it shadedxyy = 2x + 1(0;0)
  6. State the answer in words as well as on the graph.all points on or below the line y=2x+1\text{all points on or below the line } y = 2x + 1

Practice

Boundary line, solid or dashed, test a point.

1
Solve 3x−4≤11\text{Solve } 3x - 4 \leq 11
Answer3x≤15∴x≤53x \leq 15 \quad\therefore\quad x \leq 5
2
Solve −5x>20\text{Solve } -5x > 20
Answerx<−4(sign flips)x < -4 \quad \text{(sign flips)}
3
For y>x−3, is the boundary line solid or dashed?\text{For } y > x - 3, \text{ is the boundary line solid or dashed?}
AnswerDashed — the sign is >, so the edge is not included.\text{Dashed — the sign is } > \text{, so the edge is not included.}
4
Test (0;0) in y<−x+2. Which side is shaded?\text{Test } (0;0) \text{ in } y < -x + 2. \text{ Which side is shaded?}
Answer0<2 true∴shade the side containing the origin0 < 2 \ \text{true} \quad\therefore\quad \text{shade the side containing the origin}
5
Why can you not use (0;0) as a test point for y≥3x?\text{Why can you not use } (0;0) \text{ as a test point for } y \geq 3x?
AnswerThe line passes through the origin, so the test gives 0≥0 and decides nothing. Use another point.\text{The line passes through the origin, so the test gives } 0 \geq 0 \text{ and decides nothing. Use another point.}
6
Solve 2<x+5≤9\text{Solve } 2 < x + 5 \leq 9
Answer−3<x≤4-3 < x \leq 4
Next step
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