Hong Kong
Stage 1 - Stage 3

# Identifying Solutions to Inequalities in Two Variables

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: $y\ge2x+4$y2x+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 $y\ge2x+4$y2x+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?

1. 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.

1. Find the intercepts of the line.

$x$x-intercept $\editable{}$ $\editable{}$
2. 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
3. Sketch a graph of $y$y$\le$$-2x+2$2x+2.

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

No

A

Yes

B