# 2.7 Linear inequalities and absolute value inequalities  (Page 5/11)

 Page 5 / 11

## Verbal

When solving an inequality, explain what happened from Step 1 to Step 2:

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

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 $\text{\hspace{0.17em}}y=|x-3|$

We start by finding the x -intercept, or where the function = 0. Once we have that point, which is $\text{\hspace{0.17em}}\left(3,0\right),$ we graph to the right the straight line graph $\text{\hspace{0.17em}}y=x-3,$ and then when we draw it to the left we plot positive y values, taking the absolute value of them.

## Algebraic

For the following exercises, solve the inequality. Write your final answer in interval notation.

$4x-7\le 9$

$3x+2\ge 7x-1$

$\left(-\infty ,\frac{3}{4}\right]$

$-2x+3>x-5$

$4\left(x+3\right)\ge 2x-1$

$\left[\frac{-13}{2},\infty \right)$

$-\frac{1}{2}x\le \frac{-5}{4}+\frac{2}{5}x$

$-5\left(x-1\right)+3>3x-4-4x$

$\left(-\infty ,3\right)$

$-3\left(2x+1\right)>-2\left(x+4\right)$

$\frac{x+3}{8}-\frac{x+5}{5}\ge \frac{3}{10}$

$\left(-\infty ,-\frac{37}{3}\right]$

$\frac{x-1}{3}+\frac{x+2}{5}\le \frac{3}{5}$

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.

$|x+9|\ge -6$

All real numbers $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$|2x+3|<7$

$|3x-1|>11$

$\left(-\infty ,\frac{-10}{3}\right)\cup \left(4,\infty \right)$

$|2x+1|+1\le 6$

$|x-2|+4\ge 10$

$\left(-\infty ,-4\right]\cup \left[8,+\infty \right)$

$|-2x+7|\le 13$

$|x-7|<-4$

No solution

$|x-20|>-1$

$|\frac{x-3}{4}|<2$

$\left(-5,11\right)$

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 of 3 units from the number 9

$\left[6,12\right]$

Distance of10 units from the number 4

Distance of 11 units from the number 1

$\left[-10,12\right]$

$-4<3x+2\le 18$

$3x+1>2x-5>x-7$

$3y<5-2y<7+y$

$x+7

## Graphical

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.

$|x-1|>2$

$\left(-\infty ,-1\right)\cup \left(3,\infty \right)$

$|x+3|\ge 5$

$|x+7|\le 4$

$\left[-11,-3\right]$

$|x-2|<7$

$|x-2|<0$

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 $\text{\hspace{0.17em}}x=-3.$

$\left(-\infty ,-3\right)$

$x+1>x+4$

$\frac{1}{2}x+1>\frac{1}{2}x-5$

Where the blue line is above the orange line; always. All real numbers.

$\left(-\infty ,-\infty \right)$

$4x+1<\frac{1}{2}x+3$

## Numeric

For the following exercises, write the set in interval notation.

$\left\{x|-1

$\left(-1,3\right)$

$\left\{x|x\ge 7\right\}$

$\left\{x|x<4\right\}$

$\left(-\infty ,4\right)$

For the following exercises, write the interval in set-builder notation.

$\left(-\infty ,6\right)$

$\left\{x|x<6\right\}$

$\left(4,+\infty \right)$

$\left[-3,5\right)$

$\left\{x|-3\le x<5\right\}$

$\left[-4,1\right]\cup \left[9,\infty \right)$

For the following exercises, write the set of numbers represented on the number line in interval notation.

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