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. $2+r$

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

19. $2>\frac{a-2}{5}$

21. $-47 \geqslant 8-5 x$

23. $-2(3+k)$

25. $18$

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

29. $-r-5(r-6)$

31. $24+4 b$

33. $-5 v-5$

35. $4+2(a+5)$

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.

$\begin{gathered}\begin{array}{c}-(k-2) > -k-20 \\-k+2 > -k-20 \\\underline {+k +k}\end{array} \\\begin{array}{c}2 > -20 \\\text { true }\end{array} \\\text { All real numbers } \mathbb{R}\end{gathered}$