8.08 Systems of inequalities
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 $y
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 $y
Practice questions
Question 1
Select the inequalities that describe the shaded region.
A coordinate plane, with the x-axis from -10 to 10 and the y-axis also from -10 to 10. Two lines are drawn on the plane: a $dashed$dashed vertical line which crosses the x-axis at $\left(1,0\right)$(1,0) and a $solid$solid horizontal line which crosses the y-axis at $\left(0,3\right)$(0,3). These lines divide the coordinate plane into four regions, and the $\text{lower left}$lower left region is shaded.
$x$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.
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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