Recall that we can graph a linear inequality in two variables in the coordinate plane. Let's apply our knowledge of graphing systems of equations and graphing inequalities in order to graph a system of inequalities.
When two or more different inequalities are to be satisfied together, the solution set is a restricted area where the inequalities overlap. For example, suppose we require $yy<x+1 and also, $y>1$y>1. The required region is bounded by the lines $y=x+1$y=x+1 and $y=1$y=1 and is shown shaded in the diagram below.
The solution region is the intersection of the two half-planes defined by the inequalities.
Note that the boundaries have been drawn as dotted lines. This is because the points that satisfy $y=x+1$y=x+1 do not satisfy the strict inequality $yy<x+1. Similarly, the points that satisfy $y=1$y=1 do not satisfy $y>1$y>1. When the boundary line is part of the solution set we use the signs 'less than or equal to' $\le$≤ or 'greater than or equal to' $\ge$≥.
Practice questions
Question 1
Select the inequalities that describe the shaded region.
$x$x$\le$≤$1$1 and $y$y$<$<$3$3
A
$x$x$<$<$3$3 or $y$y$\le$≤$1$1
B
$x$x$<$<$3$3 and $y$y$\le$≤$1$1
C
$x$x$\le$≤$1$1 or $y$y$<$<$3$3
D
$x$x$<$<$1$1 or $y$y$\le$≤$3$3
E
$x$x$<$<$1$1 and $y$y$\le$≤$3$3
F
Question 2
Sketch a graph of the system of inequalities $x$x$\le$≤$5$5 and $y$y$<$<$3$3.
Loading Graph...
Question 3
Select the inequalities that describe the shaded region.
$y$y$\ge$≥$-4x-5$−4x−5 or $y$y$\ge$≥$-3$−3
A
$y$y$\ge$≥$-4x-5$−4x−5 and $y$y$\le$≤$-3$−3
B
$y$y$\le$≤$-4x-5$−4x−5 and $y$y$\ge$≥$-3$−3
C
$y$y$\le$≤$-4x-5$−4x−5 and $-\frac{5}{4}$−54$\le$≤$-3$−3
D
Outcomes
A2.3.E
Formulate systems of at least two linear inequalities in two variables
A2.3.G
Determine possible solutions in the solution set of systems of two or more linear inequalities in two variables