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- Absolute value functions
Key concepts
- The absolute value function is commonly used to measure distances between points. See
[link] .
- Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See
[link] .
- The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See
[link] .
- In an absolute value equation, an unknown variable is the input of an absolute value function.
- If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See
[link] .
- An absolute value equation may have one solution, two solutions, or no solutions. See
[link] .
- An absolute value inequality is similar to an absolute value equation but takes the form
It can be solved by determining the boundaries of the solution set and then testing which segments are in the set. See
[link] .
- Absolute value inequalities can also be solved graphically. See
[link] .
Section exercise
Verbal
How do you solve an absolute value equation?
Isolate the absolute value term so that the equation is of the form
Form one equation by setting the expression inside the absolute value symbol,
equal to the expression on the other side of the equation,
Form a second equation by setting
equal to the opposite of the expression on the other side of the equation,
Solve each equation for the variable.
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When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?
The graph of the absolute value function does not cross the
-axis, so the graph is either completely above or completely below the
-axis.
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How do you solve an absolute value inequality algebraically?
First determine the boundary points by finding the solution(s) of the equation. Use the boundary points to form possible solution intervals. Choose a test value in each interval to determine which values satisfy the inequality.
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Algebraic
For the following exercises, solve the equations below and express the answer using set notation.
For the following exercises, find the
x - and
y -intercepts of the graphs of each function.
For the following exercises, solve each inequality and write the solution in interval notation.
Graphical
For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.
For the following exercises, graph the given functions by hand.
Technology
Use a graphing utility to graph
on the viewing window
Identify the corresponding range. Show the graph.
range:
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For the following exercises, graph each function using a graphing utility. Specify the viewing window.
Extensions
For the following exercises, solve the inequality.
If possible, find all values of
such that there are no
-intercepts for
There is no solution for
that will keep the function from having a
-intercept. The absolute value function always crosses the
-intercept when
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Real-world applications
Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and
represents the distance from city B to city A, express this using absolute value notation.
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The true proportion
of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.
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Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable
for the score.
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A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using
as the diameter of the bearing, write this statement using absolute value notation.
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The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is
inches, express the tolerance using absolute value notation.
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Questions & Answers
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
what is the change in momentum of a body?
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
physics, biology and chemistry
this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you.
Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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