# 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

explain and give four Example hyperbolic function
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Abdullahi
hi mam
Mark
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1