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4. Linear Functions
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Math I

4.08 Graphing linear 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 the $xy$xy-plane

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. Determine 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. We've picked one point on either side of the line: $\left(0,0\right)$(0,0) marked in blue and $\left(-5,5\right)$(−5,5) marked in green.

Let's test the origin, $\left(0,0\right)$(0,0) first. We will use LHS to refer to the "Left Hand Side" and RHS to refer to "Right Hand Side" of the equation.

$LHS$LHS $=$= $y$y
  $=$= $0$0
$RHS$RHS $=$= $2x+4$2x+4
  $=$= $2\times0+4$2×0+4
  $=$= $4$4
$LHS$LHS $<$< $RHS$RHS
$y$y $<$< $2x+4$2x+4

The origin does not satisfy our inequality, 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 $\ge$≥ $RHS$RHS
$y$y $\ge$≥ $2x+4$2x+4

$\left(-5,5\right)$(−5,5) satisfies our inequality, so we will shade that side on our graph. 

Remember!
  • To graph less than ($<$<) or greater than ($>$>) use a dashed line, as the points on the line are NOT part of the solution set
  • To graph less than or equal to ($\le$≤) or greater than or equal to ($\ge$≥) use a solid line, as the points on the line are part of the solution set

 

Practice questions

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.

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?

    $\left(2,3\right)$(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.

    Loading Graph...

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

    No

    A

    Yes

    B

Outcomes

I.A.REI.12

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

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