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

 Page 6 / 11

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

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

## 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

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
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jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function