7.06 Systems of inequalities

Lesson

Practice

Lesson

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.

Loading Graph...

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.

Loading Graph...

Question 3

Select the inequalities that describe the shaded region.

Loading Graph...

A coordinate plane, with the x-axis from $-10$−10 to $10$10 and the y-axis also from $-10$−10 to $10$10. Two lines are drawn on the plane: a $solid$solid horizontal line which crosses the y-axis at $\left(0,-3\right)$(0,−3), and a $solid$solid line which crosses the x-axis at $\left(-\frac{5}{4},0\right)$(−54,0) and the y-axis at $\left(0,-5\right)$(0,−5). These lines intersect, dividing the coordinate plane into four regions. The $\text{upper left}$upper left region is shaded.

$y$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

I.A.CED.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. 'Linear and exponential (integer inputs only)

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.