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Solve the inequality and write the answer in interval notation: 5 6 x 3 4 + 8 3 x .

[ 3 14 , )

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Understanding compound inequalities

A compound inequality    includes two inequalities in one statement. A statement such as 4 < x 6 means 4 < x and x 6. There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.

Solving a compound inequality

Solve the compound inequality: 3 2 x + 2 < 6.

The first method is to write two separate inequalities: 3 2 x + 2 and 2 x + 2 < 6. We solve them independently.

3 2 x + 2 and 2 x + 2 < 6 1 2 x 2 x < 4 1 2 x x < 2

Then, we can rewrite the solution as a compound inequality, the same way the problem began.

1 2 x < 2

In interval notation, the solution is written as [ 1 2 , 2 ) .

The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.

3 2 x + 2 < 6 1 2 x < 4 Isolate the variable term, and subtract 2 from all three parts . 1 2 x < 2 Divide through all three parts by 2 .

We get the same solution: [ 1 2 , 2 ) .

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Solve the compound inequality: 4 < 2 x 8 10.

6 < x 9 or ( 6 , 9 ]

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Solving a compound inequality with the variable in all three parts

Solve the compound inequality with variables in all three parts: 3 + x > 7 x 2 > 5 x 10.

Let's try the first method. Write two inequalities :

3 + x > 7 x 2 and 7 x 2 > 5 x 10 3 > 6 x 2 2 x 2 > −10 5 > 6 x 2 x > −8 5 6 > x x > −4 x < 5 6 −4 < x

The solution set is −4 < x < 5 6 or in interval notation ( −4 , 5 6 ) . Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line. See [link] .

A number line with the points -4 and 5/6 labeled.  Dots appear at these points and a line connects these two dots.
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Solve the compound inequality: 3 y < 4 5 y < 5 + 3 y .

( 1 8 , 1 2 )

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Solving absolute value inequalities

As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at ( x , 0 ) has an absolute value of x , as it is x units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.

An absolute value inequality is an equation of the form

| A | < B , | A | B , | A | > B , or  | A | B ,

Where A , and sometimes B , represents an algebraic expression dependent on a variable x. Solving the inequality means finding the set of all x - values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.

There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.

Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of x- values such that the distance between x and 600 is less than 200. We represent the distance between x and 600 as | x 600 | , and therefore, | x 600 | 200 or

Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
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Dominic
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Lukman Reply
_3_2_1
felecia
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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
SABAL Reply
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
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Ifeanyi
on number 2 question How did you got 2x +2
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x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
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?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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corri
X=16
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16
Makan
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4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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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
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x+2y-z=7
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-1
Shedrak
Practice Key Terms 4

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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