<< 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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
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