Linear Equations

UK Secondary (7-11)

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.

Let's look at this process using an example: $y\ge2x+4$`y`≥2`x`+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`≥2`x`+4, we're going to use a solid line to draw the line $y=2x+4$`y`=2`x`+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

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

No

AYes

BNo

AYes

B

Write the inequality that describes the points in the shaded region.

Loading Graph...

Consider the line $y=-2x+2$`y`=−2`x`+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`$\le$≤ $-2x+2$−2`x`+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$\left(2,3\right)$(2,3)

A$\left(3,-6\right)$(3,−6)

B$\left(4,-2\right)$(4,−2)

C$\left(1,2\right)$(1,2)

DSketch a graph of $y$

`y`$\le$≤$-2x+2$−2`x`+2.Loading Graph...Do the points on the line satisfy the inequality $y$

`y`$\le$≤ $-2x+2$−2`x`+2?No

AYes

BNo

AYes

B