Graphing a linear inequality

What you will be able to do

• solve a linear inequality in one variable and show it on a number line;
• draw the boundary line of an inequality in two variables;
• choose the side to shade with a test point;
• tell a solid boundary from a dashed one.

From one variable to two

You know that $2x + 3 = 11$ has one answer, $x = 4$. The inequality $2x + 3 \le 11$ has many answers: every number up to $4$. Solve it like an equation. Take $3$ from both sides to get $2x \le 8$. Divide by $2$ to get $x \le 4$.

One rule is new. When you multiply or divide both sides by a negative number, the sign turns round. So $-2x \le 8$ gives $x \ge -4$.

On a number line a filled dot at $4$ means that $4$ is included. An open dot means it is not, as in $x < 4$.

Now take two variables. You have $12$ rupees. A pen costs $3$ rupees and a pencil costs $2$ rupees. If you buy $x$ pens and $y$ pencils, you spend $3x + 2y$ rupees. You can afford the purchase when

Every pair $(x, y)$ is a point on the graph. The question is: which points can you afford?

Draw the boundary, then test a point

Follow four steps.

1. Draw the boundary line. Change $\le$ to $=$. The line $3x + 2y = 12$ is easiest to draw from its intercepts. Put $y = 0$ to get $x = 4$. Put $x = 0$ to get $y = 6$. Join $(4, 0)$ and $(0, 6)$.
2. Test one point. Try the origin $(0, 0)$, which is not on the line. Then $3(0) + 2(0) = 0$, and $0 \le 12$ is true.
3. Shade the side that holds the test point when the test is true. If it is false, shade the other side.
4. Choose the line style. The signs $\le$ and $\ge$ include the line, so draw it solid. The signs $<$ and $>$ leave it out, so draw it dashed.

The point $(4, 3)$ gives $3(4) + 2(3) = 18$. This is more than $12$, so it lies outside the shading. The shop would want $18$ rupees.

A worked example with a dashed line

Draw the region $2x - y > 4$.

1. The boundary. Draw $2x - y = 4$. Put $y = 0$ to get $x = 2$. Put $x = 0$ to get $y = -4$. The sign is $>$, so the line is dashed.
2. The test. At the origin, $2(0) - 0 = 0$, and $0 > 4$ is false. So shade the side that does not hold the origin.
3. Check. Try $(4, 0)$, which is on the shaded side. Then $2(4) - 0 = 8$, and $8 > 4$ is true.

Three special cases

• Two inequalities you will always meet. $x \ge 0$ is the region to the right of the $y$ axis, and $y \ge 0$ is the region above the $x$ axis. Together they give the first quadrant. A number of pens or hours cannot be negative, so these two appear in every problem of this chapter.
• One variable missing. $x \le 5$ is a vertical line with the shading to its left. $y \ge 2$ is a horizontal line with the shading above it.
• A line through the origin. For $y \ge 2x$ the origin lies on the line, so it tells you nothing. Test another point, such as $(0, 1)$.

Some textbooks shade the unwanted side and leave the wanted region white. Check which way your book does it. We shade the wanted region, as NCERT does.

Practise with new numbers

Try it in Python

The program tests six points against $3x + 2y \le 12$. Then it shades the allowed side and colours each point green if it is allowed and red if it is not.

Common mistakes

• Shading without a test. Do not guess the side. One test point settles it.
• Testing a point on the line. The origin is no use when the line passes through it.
• A solid line for a strict sign. Strict signs, $<$ and $>$, need a dashed line.
• Forgetting to turn the sign. Dividing by $-2$ turns $\le$ into $\ge$.

Remember these

• The boundary line is the equation with the sign replaced by $=$. Draw it from its intercepts.
• Test a point that is not on the line. Shade its side if the test is true, and the other side if it is false.
• Solid for $\le$ and $\ge$. Dashed for $<$ and $>$.
• $x \ge 0$ and $y \ge 0$ together mean the first quadrant.

Where this is used

The SAT asks whether a point is a solution of an inequality or a system, and the test above answers it: put in the numbers. In India, graphing inequalities is the first step of the Class 12 chapter on linear programming. The same topic appears in A-Level Decision Mathematics on some exam boards.

References

• NCERT, Mathematics Textbook for Class XI, the chapter on Linear Inequalities (graphical solution of linear inequalities in two variables).
• NCERT, Mathematics Textbook for Class XII, Part II, Chapter 12, Linear Programming (section on the graphical method).

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Solve $3 - 2x \ge 9$.

2. Is the point $(2, 5)$ a solution of $4x + y \le 12$?

3. Where does the line $5x + 2y = 20$ cut the two axes?

4. You draw $x - 2y > 6$. Which side do you shade, and is the line solid or dashed?

5. For $y \ge 3x$, why is the origin a poor test point? Which point would you use?