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
NC.M1.A-CED.3
Create systems of linear equations and inequalities to model situations in context.
NC.M1.A-REI.12
Represent the solutions of a linear inequality or a system of linear inequalities graphically as a region of the plane.