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$2`x`+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$2`x`+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.

Loading Graph...

A line is plotted on a Cartesian coordinate plane. The line is solid and is horizontal to the $x$x-axis where it intercepts $y$y-axis at $5$5. The region below the line is shaded.

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?
$No$
A
$Yes$
B

$x$`x`-intercept	$\editable{}$
$y$`y`-intercept	$\editable{}$

Outcomes

NA6-5

Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns

91027

Apply algebraic procedures in solving problems

Identifying Solutions to Inequalities in Two Variables