<< Chapter < Page Chapter >> Page >

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

[ 3 14 , )

Got questions? Get instant answers now!

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 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

6 < x 9 or ( 6 , 9 ]

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve the compound inequality: 3 y < 4 5 y < 5 + 3 y .

( 1 8 , 1 2 )

Got questions? Get instant answers now!

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

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask