# 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? ## Chapter review exercises ## 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 if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero Rahul Reply sin^4+sin^2=1, prove that tan^2-tan^4+1=0 SAYANTANI Reply what is the formula used for this question? "Jamal wants to save$54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
what is sine?
what is the standard form of 1
1×10^0
Akugry
Evalute exponential functions
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad