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When solving an inequality, explain what happened from Step 1 to Step 2:
$\begin{array}{ll}\text{Step1}\hfill & \phantom{\rule{2em}{0ex}}-2x6\hfill \\ \text{Step2}\hfill & \phantom{\rule{3em}{0ex}}x-3\hfill \end{array}$
When we divide both sides by a negative it changes the sign of both sides so the sense of the inequality sign changes.
When solving an inequality, we arrive at:
$\begin{array}{l}x+2<x+3\hfill \\ \phantom{\rule{1.2em}{0ex}}2<3\hfill \end{array}$
Explain what our solution set is.
When writing our solution in interval notation, how do we represent all the real numbers?
$\left(-\infty ,\infty \right)$
When solving an inequality, we arrive at:
$\begin{array}{l}x+2>x+3\hfill \\ \phantom{\rule{1.2em}{0ex}}2>3\hfill \end{array}$
Explain what our solution set is.
Describe how to graph $\text{\hspace{0.17em}}y=\left|x-3\right|$
We start by finding the x -intercept, or where the function = 0. Once we have that point, which is $\text{\hspace{0.17em}}(3,0),$ we graph to the right the straight line graph $\text{\hspace{0.17em}}y=x\mathrm{-3},$ and then when we draw it to the left we plot positive y values, taking the absolute value of them.
For the following exercises, solve the inequality. Write your final answer in interval notation.
$4x-7\le 9$
$\mathrm{-2}x+3>x-5$
$-\frac{1}{2}x\le \frac{-5}{4}+\frac{2}{5}x$
$\mathrm{-3}(2x+1)>\mathrm{-2}(x+4)$
$\frac{x+3}{8}-\frac{x+5}{5}\ge \frac{3}{10}$
$\left(-\infty ,-\frac{37}{3}\right]$
$\frac{x-1}{3}+\frac{x+2}{5}\le \frac{3}{5}$
For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.
$\left|x+9\right|\ge \mathrm{-6}$
All real numbers $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$
$\left|2x+3\right|<7$
$|3x-1|>11$
$\left(-\infty ,\frac{-10}{3}\right)\cup \left(4,\infty \right)$
$\left|2x+1\right|+1\le 6$
$\left|x-2\right|+4\ge 10$
$\left(-\infty ,\mathrm{-4}\right]\cup \left[8,+\infty \right)$
$\left|\mathrm{-2}x+7\right|\le 13$
$|x-20|>\mathrm{-1}$
$\left|\frac{x-3}{4}\right|<2$
$\left(\mathrm{-5},11\right)$
For the following exercises, describe all the x -values within or including a distance of the given values.
Distance of 5 units from the number 7
Distance of10 units from the number 4
Distance of 11 units from the number 1
$\left[\mathrm{-10},12\right]$
For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.
$\mathrm{-4}<3x+2\le 18$
$3x+1>2x-5>x-7$
$\begin{array}{ll}x>-6\text{and}x-2\hfill & \phantom{\rule{2em}{0ex}}\text{Taketheintersectionoftwosets}.\hfill \\ x-2,\text{\hspace{1em}}(-2,+\infty )\hfill & \hfill \end{array}$
$3y<5-2y<7+y$
$2x-5<\mathrm{-11}\text{or}5x+1\ge 6$
$\begin{array}{ll}x<-3\text{\hspace{1em}}\mathrm{or}\text{\hspace{1em}}x\ge 1\hfill & \phantom{\rule{2em}{0ex}}\text{Taketheunionofthetwosets}.\hfill \\ (-\infty ,-3){{{\displaystyle \cup}}^{\text{}}}^{\text{}}[1,\infty )\hfill & \hfill \end{array}$
$x+7<x+2$
For the following exercises, graph the function. Observe the points of intersection and shade the x -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.
$|x-1|>2$
$\left(-\infty ,\mathrm{-1}\right)\cup \left(3,\infty \right)$
$\left|x+3\right|\ge 5$
$\left|x+7\right|\le 4$
$\left[\mathrm{-11},\mathrm{-3}\right]$
$\left|x-2\right|<7$
$\left|x-2\right|<0$
It is never less than zero. No solution.
For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y -values of the lines.
$x+3<3x-4$
$x-2>2x+1$
Where the blue line is above the orange line; point of intersection is $\text{\hspace{0.17em}}x=-3.$
$\left(-\infty ,\mathrm{-3}\right)$
$x+1>x+4$
$\frac{1}{2}x+1>\frac{1}{2}x-5$
Where the blue line is above the orange line; always. All real numbers.
$(-\infty ,-\infty )$
$4x+1<\frac{1}{2}x+3$
For the following exercises, write the set in interval notation.
$\{x|x\ge 7\}$
$\{\text{\hspace{0.17em}}x|\text{\hspace{0.17em}}x\text{isallrealnumbers}\}$
For the following exercises, write the interval in set-builder notation.
$\left(4,+\infty \right)$
$[\mathrm{-4},1]\cup [9,\infty )$
For the following exercises, write the set of numbers represented on the number line in interval notation.
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