Practice Makes Perfect
Verify Solutions to an Inequality in Two Variables
In the following exercises, determine whether each ordered pair is a solution to the given inequality.
Exercise \(\PageIndex{1}\)
Determine whether each ordered pair is a solution to the inequality \(y>x−1\):
- \((0,1)\)
- \((−4,−1)\)
- \((4,2)\)
- \((3,0)\)
- \((−2,−3)\)
Exercise \(\PageIndex{2}\)
Determine whether each ordered pair is a solution to the inequality \(y>x−3\):
- \((0,0)\)
- \((2,1)\)
- \((−1,−5)\)
- \((−6,−3)\)
- \((1,0)\)
- Answer
-
- yes
- no
- no
- yes
- no
Exercise \(\PageIndex{3}\)
Determine whether each ordered pair is a solution to the inequality \(y<x+2\):
- \((0,3)\)
- \((−3,−2)\)
- \((−2,0)\)
- \((0,0)\)
- \((−1,4)\)
Exercise \(\PageIndex{4}\)
Determine whether each ordered pair is a solution to the inequality \(y<x+5\):
- \((−3,0)\)
- \((1,6)\)
- \((−6,−2)\)
- \((0,1)\)
- \((5,−4)\)
- Answer
-
- yes
- no
- no
- yes
- yes
Exercise \(\PageIndex{5}\)
Determine whether each ordered pair is a solution to the inequality \(x+y>4\):
- \((5,1)\)
- \((−2,6)\)
- \((3,2)\)
- \((10,−5)\)
- \((0,0)\)
Exercise \(\PageIndex{6}\)
Determine whether each ordered pair is a solution to the inequality \(x+y>2\):
- \((1,1)\)
- \((4,−3)\)
- \((0,0)\)
- \((−8,12)\)
- \((3,0)\)
- Answer
-
- no
- no
- no
- yes
- yes
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Exercise \(\PageIndex{7}\)
Write the inequality shown by the graph with the boundary line \(y=3x−4\).
-

Exercise \(\PageIndex{8}\)
Write the inequality shown by the graph with the boundary line \(y=2x−4\).

- Answer
-
\(y<2 x-4\)
Exercise \(\PageIndex{9}\)
Write the inequality shown by the graph with the boundary line \(y=\frac{1}{2} x+1\)

Exercise \(\PageIndex{10}\)
Write the inequality shown by the graph with the boundary line \(y=-\frac{1}{3} x-2\)

- Answer
-
\(y \leq-\frac{1}{3} x-2\)
Exercise \(\PageIndex{11}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(x+y=5\).

Exercise \(\PageIndex{12}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(x+y=3\).

- Answer
-
\(x+y \geq 3\)
Exercise \(\PageIndex{13}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(2x+y=−4\).
-

Exercise \(\PageIndex{14}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(x+2y=−2\).

- Answer
-
\(x+2 y \geq-2\)
Exercise \(\PageIndex{15}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(3x−y=6\).
-

Exercise \(\PageIndex{16}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(2x−y=4\).

- Answer
-
\(2 x-y<4\)
Exercise \(\PageIndex{17}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(2x−5y=10\).
-

Exercise \(\PageIndex{18}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(4x−3y=12\).

- Answer
-
\(4 x-3 y>12\)
Graph Linear Inequalities
In the following exercises, graph each linear inequality.
Exercise \(\PageIndex{19}\)
Graph the linear inequality \(y>\frac{2}{3} x-1\)
Exercise \(\PageIndex{20}\)
Graph the linear inequality \(y<\frac{3}{5} x+2\)
- Answer
-

Exercise \(\PageIndex{21}\)
Graph the linear inequality \(y \leq-\frac{1}{2} x+4\)
Exercise \(\PageIndex{22}\)
Graph the linear inequality \(y \geq-\frac{1}{3} x-2\)
- Answer
-

Exercise \(\PageIndex{23}\)
Graph the linear inequality \(x-y \leq 3\)
Exercise \(\PageIndex{24}\)
Graph the linear inequality \(x-y \geq -2\)
- Answer
-

Exercise \(\PageIndex{25}\)
Graph the linear inequality \(4x+y>-4\)
Exercise \(\PageIndex{26}\)
Graph the linear inequality \(x+5y<-5\)
- Answer
-

Exercise \(\PageIndex{27}\)
Graph the linear inequality \(3 x+2 y \geq-6\)
Exercise \(\PageIndex{28}\)
Graph the linear inequality \(4 x+2 y \geq-8\)
- Answer
-

Exercise \(\PageIndex{29}\)
Graph the linear inequality \(y>4x\)
Exercise \(\PageIndex{30}\)
Graph the linear inequality \(y>x\)
- Answer
-

Exercise \(\PageIndex{31}\)
Graph the linear inequality \(y \leq-x\)
Exercise \(\PageIndex{32}\)
Graph the linear inequality \(y \leq-3 x\)
- Answer
-

Exercise \(\PageIndex{33}\)
Graph the linear inequality \(y \geq-2\)
Exercise \(\PageIndex{34}\)
Graph the linear inequality \(y<-1\)
- Answer
-

Exercise \(\PageIndex{35}\)
Graph the linear inequality \(y<4\)
Exercise \(\PageIndex{36}\)
Graph the linear inequality \(y \geq 2\)
- Answer
-

Exercise \(\PageIndex{37}\)
Graph the linear inequality \(x \leq 5\)
Exercise \(\PageIndex{38}\)
Graph the linear inequality \(x>-2\)
- Answer
-

Exercise \(\PageIndex{39}\)
Graph the linear inequality \(x>-3\)
Exercise \(\PageIndex{40}\)
Graph the linear inequality \(x \leq 4\)
- Answer
-

Exercise \(\PageIndex{41}\)
Graph the linear inequality \(x-y<4\)
Exercise \(\PageIndex{42}\)
Graph the linear inequality \(x-y<-3\)
- Answer
-

Exercise \(\PageIndex{43}\)
Graph the linear inequality \(y \geq \frac{3}{2} x\)
Exercise \(\PageIndex{44}\)
Graph the linear inequality \(y \leq \frac{5}{4} x\)
- Answer
-

Exercise \(\PageIndex{45}\)
Graph the linear inequality \(y>-2 x+1\)
Exercise \(\PageIndex{46}\)
Graph the linear inequality \(y<-3 x-4\)
- Answer
-

Exercise \(\PageIndex{47}\)
Graph the linear inequality \(x \leq-1\)
Exercise \(\PageIndex{48}\)
Graph the linear inequality \(x \geq 0\)
- Answer
-
