Finding Regions from Multiple Inequalities

Worked example

1. 1

   $y\le 2x+4$ — The inequality. Its line is y = 2x + 4, and we need to know which side of it to keep.

2. 2

   $0\le 2(0)+4$ — Put in x = 0 and y = 0. (0, 0) is not on the line, so it sits clearly on one side.

3. 3

   $0\le 4$ — Work out the right side: 2 × 0 + 4 = 4.

4. 4

   $\text{true}✓$ — The statement is true, so (0, 0) is on the side you want. If it were false, you would want the other side.

Worked example

1. 1

   $x=-1,y=3,y=x-2$ — Name the three boundary lines first. Each answer is one of these with = swapped for a sign.

2. 2

   $(0,0)\text{is inside R}$ — Pick a point inside R, not on any line. It will tell you every sign.

3. 3

   $0\ge -1\Rightarrow x\ge -1$ — For x = −1: 0 is bigger than −1, so R is where x is bigger. The line is solid, so use ≥, not >.

4. 4

   $0<3\Rightarrow y<3$ — For y = 3: 0 is smaller than 3. This line is dashed, so use < because points on it are not in R.

5. 5

   $0\text{vs}0-2=-2$ — For y = x − 2: put the point into both sides. The right side is 0 − 2 = −2.

6. 6

   $0\ge -2\Rightarrow y\ge x-2$ — 0 is bigger than −2, so R is where y is bigger. Solid line, so ≥.

7. 7

   $x\ge -1,y<3,y\ge x-2$ — The answer: three inequalities, one for each line.

Worked example

1. 1

   $(0,5)\text{and}(2,1)$ — Read two points where the line crosses grid corners exactly. Exact points give an exact gradient.

2. 2

   $m=\frac{1-5}{2-0}$ — Gradient = change in y ÷ change in x, going from the first point to the second.

3. 3

   $m=\frac{-4}{2}=-2$ — Going right 2 drops 4. A line that slopes down from left to right always has a negative gradient.

4. 4

   $c=5$ — It crosses the y-axis at (0, 5), so the y-intercept is 5.

5. 5

   $y=-2x+5$ — Put m and c into y = mx + c. This is the missing line.

6. 6

   $0\le -2(0)+5$ — Now pick the sign. (0, 0) is inside R, so test it: the right side is 5.

7. 7

   $y\le -2x+5$ — 0 ≤ 5 is true and the line is solid, so ≤. The form 2x + y ≤ 5 means the same and also scores.

Worked example

1. 1

   $x=-2,x=3,y=0,y=4$ — Four lines: two vertical, two horizontal. R is the rectangle between them.

2. 2

   $-2\le x$ — R is to the right of the solid line x = −2, so x is at least −2.

3. 3

   $x<3$ — R is to the left of x = 3. That line is dashed, so x stays below 3 and never equals it.

4. 4

   $-2\le x<3$ — Join the two facts about x. x sits between the two vertical lines.

5. 5

   $0\le y\le 4$ — Do the same for y: above y = 0 and below y = 4. Both lines are solid, so both signs are ≤.

Worked example

1. 1

   $x>1,y\ge 1,x+2y\le 9$ — The question. Three inequalities, so three lines to draw.

2. 2

   $x=1\text{(dashed)}$ — Swap > for = to get the line: vertical, through 1 on the x-axis. Dashed, because > does not include the line.

3. 3

   $y=1\text{(solid)}$ — Horizontal, through 1 on the y-axis. Solid, because ≥ includes the line.

4. 4

   $x=0:2y=9,y=4.5$ — For x + 2y = 9, find where it meets the y-axis: put x = 0, then divide by 2.

5. 5

   $y=0:x=9$ — Put y = 0 to find where it meets the x-axis. Join (0, 4.5) to (9, 0) with a ruler: solid, because ≤.

6. 6

   $0+2(0)=0\le 9$ — Test (0, 0) in the sloping one: true, so keep the side with the origin and shade away the other side.

7. 7

   $\text{shade}x\le 1\text{and}y<1$ — x > 1 keeps the right of x = 1, so shade its left. y ≥ 1 keeps above y = 1, so shade below.

8. 8

   $(1,1),(1,4),(7,1)$ — The clear triangle with these corners is R. Write R inside it.

Worked example

1. 1

   $x\ge 1,y\ge 1,x+y<5$ — The region. List every point (x, y) in R where x and y are whole numbers.

2. 2

   $x=1:1+y<5$ — Start with the smallest x allowed, x = 1, and put it into x + y < 5.

3. 3

   $y<4\Rightarrow y=1,2,3$ — Take 1 from both sides. y must be a whole number, at least 1 and below 4.

4. 4

   $x=2:y<3\Rightarrow y=1,2$ — Next column. 2 + y < 5, so y < 3.

5. 5

   $x=3:y<2\Rightarrow y=1$ — Next column. 3 + y < 5, so y < 2. Only y = 1 works.

6. 6

   $x=4:y<1$ — No whole number is at least 1 and below 1, so this column is empty. Stop here.

7. 7

   $(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)$ — Six points. (1, 4) is not one of them: it is on the dashed line x + y = 5, which is not in R.