Identifying Solutions to Inequalities in Two Variables

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.

 

Examples

Question 1

Question 2

Question 3