Identifying Solutions to Inequalities in Two Variables

Lesson

Practice

Lesson

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