# 2.7 Linear inequalities and absolute value inequalities  (Page 6/11)

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$\left(-2,1\right]$

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

## Technology

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter y2 = the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall (2 nd CALC 5:intersection, 1 st curve, enter, 2 nd curve, enter, guess, enter). Copy a sketch of the graph and shade the x -axis for your solution set to the inequality. Write final answers in interval notation.

$|x+2|-5<2$

$\frac{-1}{2}|x+2|<4$

Where the blue is below the orange; always. All real numbers. $\text{\hspace{0.17em}}\left(-\infty ,+\infty \right).$

$|4x+1|-3>2$

$|x-4|<3$

Where the blue is below the orange; $\text{\hspace{0.17em}}\left(1,7\right).$

$|x+2|\ge 5$

## Extensions

Solve $\text{\hspace{0.17em}}|3x+1|=|2x+3|$

$x=2,\frac{-4}{5}$

Solve ${x}^{2}-x>12$

$\frac{x-5}{x+7}\le 0,$ $x\ne -7$

$\left(-7,5\right]$

$p=-{x}^{2}+130x-3000\text{\hspace{0.17em}}$ is a profit formula for a small business. Find the set of x -values that will keep this profit positive.

## Real-world applications

In chemistry the volume for a certain gas is given by $\text{\hspace{0.17em}}V=20T,$ where V is measured in cc and T is temperature in ºC. If the temperature varies between 80ºC and 120ºC, find the set of volume values.

$\begin{array}{l}80\le T\le 120\\ 1,600\le 20T\le 2,400\end{array}$

A basic cellular package costs $20/mo. for 60 min of calling, with an additional charge of$.30/min beyond that time.. The cost formula would be $\text{\hspace{0.17em}}C=\text{}20+.30\left(x-60\right).\text{\hspace{0.17em}}$ If you have to keep your bill lower than \$50, what is the maximum calling minutes you can use?

## The Rectangular Coordinate Systems and Graphs

For the following exercises, find the x -intercept and the y -intercept without graphing.

$4x-3y=12$

x -intercept: $\text{\hspace{0.17em}}\left(3,0\right);$ y -intercept: $\text{\hspace{0.17em}}\left(0,-4\right)$

$2y-4=3x$

For the following exercises, solve for y in terms of x , putting the equation in slope–intercept form.

$5x=3y-12$

$y=\frac{5}{3}x+4$

$2x-5y=7$

For the following exercises, find the distance between the two points.

$\left(-2,5\right)\left(4,-1\right)$

$\sqrt{72}=6\sqrt{2}$

$\left(-12,-3\right)\left(-1,5\right)$

Find the distance between the two points $\text{\hspace{0.17em}}\left(-71,432\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{(511,218)}\text{\hspace{0.17em}}$ using your calculator, and round your answer to the nearest thousandth.

$620.097$

For the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

midpoint is $\text{\hspace{0.17em}}\left(2,\frac{23}{2}\right)$

For the following exercises, construct a table and graph the equation by plotting at least three points.

$y=\frac{1}{2}x+4$

$4x-3y=6$

 x y 0 −2 3 2 6 6

## Linear Equations in One Variable

For the following exercises, solve for $\text{\hspace{0.17em}}x.$

$5x+2=7x-8$

$3\left(x+2\right)-10=x+4$

$x=4$

$7x-3=5$

$12-5\left(x+1\right)=2x-5$

$x=\frac{12}{7}$

$\frac{2x}{3}-\frac{3}{4}=\frac{x}{6}+\frac{21}{4}$

For the following exercises, solve for $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ State all x -values that are excluded from the solution set.

$\frac{x}{{x}^{2}-9}+\frac{4}{x+3}=\frac{3}{{x}^{2}-9}\text{\hspace{0.17em}}$ $x\ne 3,-3$

No solution

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

For the following exercises, find the equation of the line using the point-slope formula.

Passes through these two points: $\text{\hspace{0.17em}}\left(-2,1\right)\text{,}\left(4,2\right).$

$y=\frac{1}{6}x+\frac{4}{3}$

Passes through the point $\text{\hspace{0.17em}}\left(-3,4\right)\text{\hspace{0.17em}}$ and has a slope of $\text{\hspace{0.17em}}\frac{-1}{3}.$

Passes through the point $\text{\hspace{0.17em}}\left(-3,4\right)\text{\hspace{0.17em}}$ and is parallel to the graph $\text{\hspace{0.17em}}y=\frac{2}{3}x+5.$

$y=\frac{2}{3}x+6$

Passes through these two points: $\text{\hspace{0.17em}}\left(5,1\right)\text{,}\left(5,7\right).$

## Models and Applications

For the following exercises, write and solve an equation to answer each question.

The number of males in the classroom is five more than three times the number of females. If the total number of students is 73, how many of each gender are in the class?

females 17, males 56

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
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