# 2.7 Linear inequalities and absolute value inequalities  (Page 7/11)

 Page 7 / 11

A man has 72 ft. of fencing to put around a rectangular garden. If the length is 3 times the width, find the dimensions of his garden.

A truck rental is $25 plus$.30/mi. Find out how many miles Ken traveled if his bill was \$50.20.

84 mi

## Complex Numbers

For the following exercises, use the quadratic equation to solve.

${x}^{2}-5x+9=0$

$2{x}^{2}+3x+7=0$

$x=\frac{-3}{4}±\frac{i\sqrt{47}}{4}$

For the following exercises, name the horizontal component and the vertical component.

$4-3i$

$-2-i$

horizontal component $\text{\hspace{0.17em}}-2;$ vertical component $\text{\hspace{0.17em}}-1$

For the following exercises, perform the operations indicated.

$\left(9-i\right)-\left(4-7i\right)$

$\left(2+3i\right)-\left(-5-8i\right)$

$7+11i$

$2\sqrt{-75}+3\sqrt{25}$

$\sqrt{-16}+4\sqrt{-9}$

$16i$

$-6i\left(i-5\right)$

${\left(3-5i\right)}^{2}$

$-16-30i$

$\sqrt{-4}·\sqrt{-12}$

$\sqrt{-2}\left(\sqrt{-8}-\sqrt{5}\right)$

$-4-i\sqrt{10}$

$\frac{2}{5-3i}$

$\frac{3+7i}{i}$

$x=7-3i$

For the following exercises, solve the quadratic equation by factoring.

$2{x}^{2}-7x-4=0$

$3{x}^{2}+18x+15=0$

$x=-1,-5$

$x=0,\frac{9}{7}$

For the following exercises, solve the quadratic equation by using the square-root property.

${x}^{2}=49$

${\left(x-4\right)}^{2}=36$

$x=10,-2$

For the following exercises, solve the quadratic equation by completing the square.

${x}^{2}+8x-5=0$

$4{x}^{2}+2x-1=0$

$x=\frac{-1±\sqrt{5}}{4}$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No real solution .

$2{x}^{2}-5x+1=0$

$15{x}^{2}-x-2=0$

$x=\frac{2}{5},\frac{-1}{3}$

For the following exercises, solve the quadratic equation by the method of your choice.

${\left(x-2\right)}^{2}=16$

${x}^{2}=10x+3$

$x=5±2\sqrt{7}$

## Other Types of Equations

For the following exercises, solve the equations.

${x}^{\frac{3}{2}}=27$

${x}^{\frac{1}{2}}-4{x}^{\frac{1}{4}}=0$

$x=0,256$

$4{x}^{3}+8{x}^{2}-9x-18=0$

$3{x}^{5}-6{x}^{3}=0$

$x=0,±\sqrt{2}$

$\sqrt{x+9}=x-3$

$\sqrt{3x+7}+\sqrt{x+2}=1$

$x=-2$

$|3x-7|=5$

$|2x+3|-5=9$

$x=\frac{11}{2},\frac{-17}{2}$

## Linear Inequalities and Absolute Value Inequalities

For the following exercises, solve the inequality. Write your final answer in interval notation.

$5x-8\le 12$

$-2x+5>x-7$

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

$\frac{x-1}{3}+\frac{x+2}{5}\le \frac{3}{5}$

$|3x+2|+1\le 9$

$\left[\frac{-10}{3},2\right]$

$|5x-1|>14$

$|x-3|<-4$

No solution

For the following exercises, solve the compound inequality. Write your answer in interval notation.

$-4<3x+2\le 18$

$3y<1-2y<5+y$

$\left(-\frac{4}{3},\frac{1}{5}\right)$

For the following exercises, graph as described.

Graph the absolute value function and graph the constant function. Observe the points of intersection and shade the x -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.

$|x+3|\ge 5$

Graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y -values of the lines. See the interval where the inequality is true.

$x+3<3x-4$

Where the blue is below the orange line; point of intersection is $\text{\hspace{0.17em}}x=3.5.$

$\left(3.5,\infty \right)$

## Chapter practice test

Graph the following: $\text{\hspace{0.17em}}2y=3x+4.$

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

x y
0 2
2 5
4 8

Find the x- and y -intercepts for the following:

$2x-5y=6$

Find the x- and y -intercepts of this equation, and sketch the graph of the line using just the intercepts plotted.

$3x-4y=12$

$\left(0,-3\right)$ $\left(4,0\right)$

Find the exact distance between $\text{\hspace{0.17em}}\left(5,-3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,8\right).\text{\hspace{0.17em}}$ Find the coordinates of the midpoint of the line segment joining the two points.

Write the interval notation for the set of numbers represented by $\text{\hspace{0.17em}}\left\{x|x\le 9\right\}.$

$\left(-\infty ,9\right]$

Solve for x : $\text{\hspace{0.17em}}5x+8=3x-10.$

Solve for x : $\text{\hspace{0.17em}}3\left(2x-5\right)-3\left(x-7\right)=2x-9.$

$x=-15$

Solve for x : $\text{\hspace{0.17em}}\frac{x}{2}+1=\frac{4}{x}$

Solve for x : $\text{\hspace{0.17em}}\frac{5}{x+4}=4+\frac{3}{x-2}.$

$x\ne -4,2;$ $x=\frac{-5}{2},1$

The perimeter of a triangle is 30 in. The longest side is 2 less than 3 times the shortest side and the other side is 2 more than twice the shortest side. Find the length of each side.

$\frac{{x}^{2}}{3}-x=\frac{-1}{2}$

$x=\frac{3±\sqrt{3}}{2}$

Solve: $\text{\hspace{0.17em}}3x-8\le 4.$

Solve: $\text{\hspace{0.17em}}|2x+3|<5.$

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

Solve: $\text{\hspace{0.17em}}|3x-2|\ge 4.$

For the following exercises, find the equation of the line with the given information.

Passes through the points $\text{\hspace{0.17em}}\left(-4,2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,-3\right).$

$y=\frac{-5}{9}x-\frac{2}{9}$

Has an undefined slope and passes through the point $\text{\hspace{0.17em}}\left(4,3\right).$

Passes through the point $\text{\hspace{0.17em}}\left(2,1\right)\text{\hspace{0.17em}}$ and is perpendicular to $\text{\hspace{0.17em}}y=\frac{-2}{5}x+3.$

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

Add these complex numbers: $\text{\hspace{0.17em}}\left(3-2i\right)+\left(4-i\right).$

Simplify: $\text{\hspace{0.17em}}\sqrt{-4}+3\sqrt{-16}.$

$14i$

Multiply: $\text{\hspace{0.17em}}5i\left(5-3i\right).$

Divide: $\text{\hspace{0.17em}}\frac{4-i}{2+3i}.$

$\frac{5}{13}-\frac{14}{13}i$

Solve this quadratic equation and write the two complex roots in $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ form: $\text{\hspace{0.17em}}{x}^{2}-4x+7=0.$

Solve: $\text{\hspace{0.17em}}{\left(3x-1\right)}^{2}-1=24.$

$x=2,\frac{-4}{3}$

Solve: $\text{\hspace{0.17em}}{x}^{2}-6x=13.$

Solve: $\text{\hspace{0.17em}}4{x}^{2}-4x-1=0$

$x=\frac{1}{2}±\frac{\sqrt{2}}{2}$

Solve:

$\sqrt{x-7}=x-7$

Solve: $\text{\hspace{0.17em}}2+\sqrt{12-2x}=x$

$4$

Solve: $\text{\hspace{0.17em}}{\left(x-1\right)}^{\frac{2}{3}}=9$

For the following exercises, find the real solutions of each equation by factoring.

$2{x}^{3}-{x}^{2}-8x+4=0$

$x=\frac{1}{2},2,-2$

${\left(x+5\right)}^{2}-3\left(x+5\right)-4=0$

what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Wilson
what is algebra
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
I want to know partial fraction Decomposition.
classes of function in mathematics
divide y2_8y2+5y2/y2
wish i knew calculus to understand what's going on 🙂
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
24x^5
James
10x
Axmed
24X^5
Taieb
comment écrire les symboles de math par un clavier normal
SLIMANE