We've already been introduced to inequalities which are expressions that explain a relationship between two quantities that aren't equal. We can also solve inequalities and graph these solutions on a number plane.

Graphing inequalities on number lines

Let's look at this process using an example: $y\ge2x+4$y≥2x+4

1. Graph the line as if it was an equation. If your inequality is "$\ge$≥" or "$\le$≤", use a solid line. If your inequality is "$>$>" or "$<$<", use a dashed line. Since our inequality is $y\ge2x+4$y≥2x+4, we're going to use a solid line to draw the line $y=2x+4$y=2x+4.

2. Work out which side of the line you should shade by seeing whether a point on the number plane (that doesn't lie on the line you've drawn) satisfies the inequality. To do this, you need to see whether the substituted $x$x and $y$y values satisfy the inequality. I've picked one point on either side of the line (note: not on the line): $\left(0,0\right)$(0,0) marked in blue and $\left(-5,5\right)$(−5,5) marked in green.

Let's test he origin $\left(0,0\right)$(0,0) first:

$LHS$LHS

$=$=

$y$y

$=$=

$0$0

$RHS$RHS

$=$=

$2x+4$2x+4

$=$=

$2\times0+4$2×0+4

$=$=

$4$4

$<$<

$LHS$LHS

The origin does not satisfy our equation, so we will not shade this side of the line.

Let's check another point above the line, say $\left(-5,5\right)$(−5,5).

$LHS$LHS

$=$=

$5$5

$RHS$RHS

$=$=

$2x+4$2x+4

$=$=

$2\times\left(-5\right)+4$2×(−5)+4

$=$=

$-6$−6

$>$>

$LHS$LHS

Now that's looking good! Let's shade this side on our graph.

And that's how we do it.

Remember!

To graph "less than" ($<$<) or "more than" ($>$>), use a dashed line like so: - - - - -

To graph "less than or equal to" ($\le$≤) or "more than or equal to" ($\ge$≥), use a solid line

Examples

Question 1

Is $\left(3,2\right)$(3,2) a solution of $3x+2y$3x+2y$\ge$≥$12$12?

No

A

Yes

B

Question 2

Write the inequality that describes the points in the shaded region.

Question 3

Consider the line $y=-2x+2$y=−2x+2.

Find the intercepts of the line.

$x$x-intercept

$\editable{}$

$y$y-intercept

$\editable{}$

Which of the following points satisfies the inequality $y$y$\le$≤$-2x+2$−2x+2?

$\left(2,3\right)$(2,3)

A

$\left(3,-6\right)$(3,−6)

B

$\left(4,-2\right)$(4,−2)

C

$\left(1,2\right)$(1,2)

D

Sketch a graph of $y$y$\le$≤$-2x+2$−2x+2.

Loading Graph...

Do the points on the line satisfy the inequality $y$y$\le$≤$-2x+2$−2x+2?