Answer Key 4.1
- \((-5, \infty)\)
- \((4, \infty)\)
- \((-2, \infty)\)
- \((-\infty, 1]\)
- \((-\infty, 5]\)
- \((-5, \infty)\)
- \(x>-5\hspace{0.25in} (-5, \infty)\)
- \(x>0\hspace{0.25in} (0, \infty)\)
- \(x\ge -3 \hspace{0.25in} [-3, \infty)\)
- \(x\le 6 \hspace{0.25in} (-\infty, 6]\)
- \(x\le -1 \hspace{0.25in} (-\infty, -1]\)
- \(x < 6 \hspace{0.25in} (-\infty, 6)\)
- \(\begin{array}{rrrl}
\\ \\
(\dfrac{x}{11}&\ge &10)&(11) \\ \\
x& \ge & 110 &
\end{array}\)\(\text{Interval notation: } [110,\infty)\)
- \(\begin{array}{rrrl}
\\ \\
(-2 &\le & \dfrac{n}{13})&(13) \\ \\
-26& \le & n &
\end{array}\)\(\text{Interval notation: } [-26, \infty)\)
- \(\begin{array}{rrrlr}
\\ \\
2 &+ & r&<&3 \\
-2&&&& -2 \\
\hline
&&r&<&1
\end{array}\)\(\text{Interval notation: } (-\infty, 1)\)
- \(\begin{array}{rrrl}
\\ \\
(\dfrac{m}{5} &\le & -\dfrac{6}{5})&(5) \\ \\
m& \le & -6 &
\end{array}\)\(\text{Interval notation: } (-\infty, -6]\)
- \(\begin{array}{rrrrrr}
\\ \\ \\ \\ \\
8&+&\dfrac{n}{3}&\ge & 6 & \\
-8&&&&-8 & \\
\hline
&&(\dfrac{n}{3} &\ge & -2)& (3) \\ \\
&&n & \ge & -6 &
\end{array}\)\(\text{Interval notation: } [-6, \infty)\)
- \(\begin{array}{rrrrr}
\\ \\ \\ \\ \\
11&>&8&+ & \dfrac{x}{2} \\
-8&&-8&& \\
\hline
(3 &> & \dfrac{x}{2})& (2)& \\ \\
6 & > & x &&
\end{array}\)\(\text{Interval notation: } (-\infty, 6)\)
- \(\left(2>\dfrac{(a-2)}{5}\right)(5)\)\(\begin{array}{rrrrr}
10&>&a&-&2 \\
+2&&&+&2 \\
\hline
12&>&a&&
\end{array}\)\(\text{Interval notation: } (-\infty, 12)\)
- \(\left(\dfrac{(v-9)}{-4}\le 2 \right)(-4)\)\(\begin{array}{rrrrr}
v&-&9&\ge &-8 \\
&+&9&&+9 \\
\hline
&&v&\ge &1
\end{array}\)\(\text{Interval notation: } [1, \infty)\)
- \(\begin{array}{rrrrl}
\\ \\ \\ \\ \\
-47&\ge &8&-& 5x \\
-8&&-8&& \\
\hline
\dfrac{-55}{-5}&\ge &\dfrac{-5x}{-5} && \\ \\
11& \le & x &&
\end{array}\)\(\text{Interval notation: } [11, \infty)\)
- \(\left(\dfrac{(6+x)}{12}\le -1 \right)(12)\)\(\begin{array}{rrrrr}
6&+&x&\le &-12 \\
-6&&&&-6 \\
\hline
&&x&\le &-18
\end{array}\)\(\text{Interval notation: } (-\infty, -18]\)
- \(\begin{array}{rrrrl}
\\ \\ \\ \\ \\
\dfrac{-2}{-2}(3&+ &k)&<& \dfrac{-44}{-2} \\ \\
3&+ &k &>&22 \\
-3&&&&-3 \\
\hline
&& k&>&19
\end{array}\)\(\text{Interval notation: } (19, \infty)\)
- \(\begin{array}{rrrrr}
\\ \\ \\ \\ \\
-7n&- &10&\ge &60 \\
&+&10&&+10 \\
\hline
&&\dfrac{-7n}{-7}&\ge & \dfrac{70}{-7} \\ \\
&&n&\le &-10
\end{array}\)\(\text{Interval notation: } (-\infty, -10]\)
- \(\begin{array}{rrrrl}
\\ \\ \\ \\ \\
\dfrac{18}{-2}&<&\dfrac{-2}{-2}(-8&+&p) \\ \\
-9&>&-8&+&p \\
+8&&+8&& \\
\hline
-1&>&p&&
\end{array}\)\(\text{Interval notation: } (-\infty, -1)\)
- \(\left(5> \dfrac{x}{5}+1 \right)(5)\)\(\begin{array}{rrrrr}
25&\ge &x&+ &5 \\
-5&&&&-5 \\
\hline
20&\ge &x& &
\end{array}\)\(\text{Interval notation: } (-\infty, 20]\)
- \(\begin{array}{rrrrr}
\\ \\ \\ \\ \\
\dfrac{24}{-6}&\ge &\dfrac{-6}{-6}(m&-&6) \\ \\
-4&\le &m&-&6 \\
+6&&&+&6 \\
\hline
2&\le &m&&
\end{array}\)\(\text{Interval notation: } [2, \infty)\)
- \(\begin{array}{rrrrr}
\\ \\ \\ \\ \\
\dfrac{-8}{-8}(n&-&5)&\ge &\dfrac{0}{-8} \\ \\
n&-&5&\le &0 \\
&+&5&&+5 \\
\hline
&&n&\le &5
\end{array}\)\(\text{Interval notation: } (-\infty, 5]\)
- \(\begin{array}{rrrrrrl}
\\ \\ \\ \\ \\ \\
-r&-&5(r&-&6)&<&-18 \\
-r&-&5r&+&30&<&-18 \\
&&&-&30&&-30 \\
\hline
&&&&\dfrac{-6r}{-6}&<&\dfrac{-48}{-6} \\ \\
&&&&r&>&8 \\
\end{array}\)\(\text{Interval notation: } (8, \infty)\)
- \(\begin{array}{rrlrr}
\\ \\ \\ \\ \\ \\
\dfrac{-60}{-4}&\ge &\dfrac{-4}{-4}(-6x&-&3) \\ \\
15&\le &-6x&-&3 \\
+3&&&+&3 \\
\hline
\dfrac{18}{-6}&\le&\dfrac{-6x}{-6}&& \\ \\
-3&\ge &x &&
\end{array}\)\(\text{Interval notation: } (-\infty, -3]\)
- \(\begin{array}{rrrrrrr}
\\ \\ \\ \\ \\ \\ \\
&&\dfrac{24+4b}{4}&<&\dfrac{4}{4}(1&+&6b) \\ \\
6&+&b&<&1&+&6b \\
-1&-&b&&-1&-&b \\
\hline
&&\dfrac{5}{5}&<&\dfrac{5b}{5}&& \\ \\
&&b&>&1&&
\end{array}\)\(\text{Interval notation: } (1, \infty)\)
- \(\begin{array}{rrrrrrr}
\\ \\ \\
-8(2&-&2n)&\ge &-16&+&n \\
-16&+&16n& \ge & -16 & + & n\\
+16&-&n&&+16 &-& n\\
\hline
&&15n& \ge &0 &&\\
&&n& \ge &0 &&\\
\end{array}\)\(\text{Interval notation: } [0, \infty)\)
- \(\begin{array}{rrrrrrr}
\\ \\ \\ \\ \\ \\ \\
&&\dfrac{-5v-5}{-5}&<&\dfrac{-5}{-5}(4v&+&1) \\ \\
v&+&1&>&4v&+&1 \\
-v&-&1&&-v&-&1 \\
\hline
&&\dfrac{0}{3}&>&\dfrac{3v}{3}&& \\ \\
&&v&<&0&&
\end{array}\)\(\text{Interval notation: } (-\infty, 0)\)
- \(\begin{array}{rrrrrrrrr}
\\ \\ \\ \\ \\
-36&+&6x&>&-8(x&+&2)&+&4x \\
-36&+&6x&>&-8x&-&16&+&4x \\
+16&-&6x&&-6x&+&16&& \\
\hline
&&\dfrac{-20}{-10}&>&\dfrac{-10x}{-10}&&&& \\ \\
&&x&>&2&&&& \\
\end{array}\)\(\text{Interval notation: } (2, \infty)\)
- \(\begin{array}{rrrrrrrrl}
\\ \\ \\ \\
4&+&2(a&+&5)&<&-2(-a&-&4) \\
4&+&2a&+&10&<&2a&+&8 \\
-4&-&2a&-&10&&-2a&-&10-4 \\
\hline
&&0&<&-6&&&&
\end{array}\)\(\text{False. No solution.}\)
- \(\begin{array}{rrrrrrrrrrrrr}
\\ \\ \\ \\ \\
3(n&+&3)&+&7(8&-&8n)&<&5n&+&5&+&2 \\
3n&+&9&+&56&-&56n&<&5n&+&7&& \\
&&&&-53n&+&65&<&5n&+&7&& \\
&&&&-5n&-&65&&-5n&-&65&& \\
\hline
&&&&&&-58n&<&-58&&&& \\
&&&&&&n&>&1&&&& \\
\end{array}\)\(\text{Interval notation: } (1, \infty)\)
- \(\begin{array}{rrrrrrr}
\\ \\ \\
-(k&-&2)&>&-k&-&20 \\
-k&+&2&>&-k&-&20 \\
+k&-&2&&+k&-&2 \\
\hline
&&0&>&-22&& \\
\end{array}\)\(\text{Always true. Solution is all real numbers:} (-\infty, \infty)\)
- \(\begin{array}{rrrrrrrrl}
\\ \\ \\ \\
-(4&-&5p)&+&3&\ge &-2(8&-&5p) \\
-4&+&5p&+&3&\ge &-16&+&10p \\
&&-1&+&5p&\ge &-16&+&10p \\
&&+1&-&10p&&+1&-&10p \\
\hline
&&&&\dfrac{-5p}{-5}&\ge &\dfrac{-15}{-5}&& \\ \\
&&&&p&\le &3&& \\
\end{array}\)\(\text{Interval notation: } (-\infty, 3]\)
