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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: $y\ge2x+4$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.

  • 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



Question 1

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

  1. No




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.

  1. Find the intercepts of the line.

    $x$x-intercept $\editable{}$
    $y$y-intercept $\editable{}$
  2. Which of the following points satisfies the inequality $y$y $\le$ $-2x+2$2x+2?








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

    Loading Graph...

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






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