Practice with Solving and Graphing Inequalities

Draw a graph for each inequality and give interval notation.

1. \(n>-5\)

3. \(-2 \geqslant k\)

5. \(5 \geqslant x\)

Write an inequality for each graph.

7.

9.

11.

Solve each inequality, graph each solution, and give interval notation.

13. \(\frac{x}{11} \geqslant 10\)

15. \(2+r<3\)

17. \(8+\frac{n}{3} \geqslant 6\)

19. \(2>\frac{a-2}{5}\)

21. \(-47 \geqslant 8-5 x\)

23. \(-2(3+k)<-44\)

25. \(18<-2(-8+p)\)

27. \(24 \geqslant-6(m-6)\)

29. \(-r-5(r-6)<-18\)

31. \(24+4 b<4(1+6 b)\)

33. \(-5 v-5<-5(4 v+1)\)

35. \(4+2(a+5)<-2(-a-4)\)

37. \(-(k-2)>-k-20\)

Source: Tyler Wallace, http://www.wallace.ccfaculty.org/book/Beginning_and_Intermediate_Algebra.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. \(n > -5\)

(\(-5, 0\))

3. \(-2 \geq k\)

\(( -\infty, - 2) \)

5. \(5 \geq x\)

\( (-\infty, 5) \)

7. \(x < -2\)

9. \(x \geq 5\)

11. \(x > -2\)

13.

\(\text { (11) } \frac{x}{11} \geq 10(11)\)

\(x \geq 110\)

\(( 110, \infty )\)

15.

\(\begin{array}{r}

2+r < 3 \\

\underline {-2 \quad-2} \\

r < 1

\end{array}\)

\( (-\infty, 1)\)

17.

\(\begin{gathered}

8+\frac{n}{3} \geq 6 \\

\underline {-8 \quad-8} \\

\text { (3) } \frac{n}{3} \geq-2(3) \\

n \geq-6

\end{gathered}\)

\( (-6, \infty) \)

19.

\(\begin{aligned}

(5) 2 > \frac{a-2}{5}(5) \\

10 > a-2 \\

 \underline {+2 \quad+2} \\

12 > a

\end{aligned}\)

\( (-\infty, 12) \)

21.

\(\begin{aligned}

 -47  \geq 8-5 x \\

 \frac{-8-8}{-\frac{55}{-5}  \geq-\frac{5 x}{-5}} \\

 11  \leq x

\end{aligned}\)

\( (11, \infty) \)

23.

\(\begin{aligned}

-2(3+k) & < -44 \\

-6-2 k & < -44 \\

\underline {+6 \quad +6} \\

-\frac{2 k}{-2} & < -\frac{38}{-2} \\

k & > 19

\end{aligned}\)

\( (19, \infty) \)

25.

\(\begin{aligned}

& 18 < -2(-8+p)\\

& 18 < 16-2 p\\

& \frac{-16-16}{\frac{2}{-2} < -\frac{2 p}{-2}}\\

& -1 > p

\end{aligned}\)

\( (- \infty, -1) \)

27.

\(\begin{aligned}

&24 \geq-6(m-6)\\

&24 \geq-6 m+36\\

&\frac{-36-36}{-\frac{12}{-6} \geq-\frac{6 m}{-6}}\\

&2 \leq m

\end{aligned}\)

\( (2, \infty) \)

29.

\(\begin{aligned}

-r-5(r-6) & < -18 \\

-r-5 r+30 & < -18 \\

-6 r+30 & < -18 \\

\underline {-30 -30} \\

-\frac{6 r}{-6} & < -\frac{48}{-6} \\

r &>8

\end{aligned}\)

\( (8, \infty) \)

31.

\(\begin{aligned}

24+4 b & < 4(1+6 b) \\

24+4 b & < 4+24 b \\

\underline {-4 b -4 b} \\

24 & < 4+20 b \\

\underline {-4 -4} \\

 \frac{20}{20} & < \frac{20 b}{20} \\

1 & < b

\end{aligned}\)

\( (1, \infty) \)

33.

\(\begin{aligned}

-5 v-5 & < -5(4 v+1) \\

-5 v-5 & < -20 v-5 \\

\underline {+20 v +20 v} \\

15 v-5 & < -5 \\

\underline {+5 +5} \\

15 v & < 0 \\

v & < 0

\end{aligned}\)

\( (- \infty, 0) \)

35. 

\(\begin{gathered}

4+2(a+5) < -2(-a-4) \\

4+2 a+10 < 2 a+8 \\

14+2 a < 2 a+8 \\

\underline {-2 a-2 a} \\

14 < 8 \\

\text { false } \\

\text { No solution } \emptyset

\end{gathered}\)

37.

\(\begin{gathered}

\begin{array}{c}

-(k-2) > -k-20 \\

-k+2 > -k-20 \\

\underline {+k +k}

\end{array} \\

\begin{array}{c}

2 > -20 \\

\text { true }

\end{array} \\

\text { All real numbers } \mathbb{R}

\end{gathered}\)