# 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

#### Questions & Answers

prove sin²x+cos²x=3+cos4x
Kiddy Reply
the difference between two signed numbers is -8.if the minued is 5,what is the subtrahend
jeramie Reply
the difference between two signed numbers is -8.if the minuend is 5.what is the subtrahend
jeramie
what are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
7%2=1, 679%2=1, 866245%2=1
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
ask
Ans
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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