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0.4 Rational expression concepts -- rational equations

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This module introduces rational expressions in equations.

Rational equations

A rational equation means that you are setting two rational expressions equal to each other. The goal is to solve for x; that is, find the x value(s) that make the equation true.

Suppose I told you that:

x 8 = 3 8 size 12{ { {x} over {8} } = { {3} over {8} } } {}

If you think about it, the x in this equation has to be a 3. That is to say, if x=3 then this equation is true; for any other x value, this equation is false.

This leads us to a very general rule.

A very general rule about rational equations

If you have a rational equation where the denominators are the same, then the numerators must be the same.

This in turn suggests a strategy: find a common denominator, and then set the numerators equal.

Example: Rational Equation
3 x 2 + 12 x + 36 = 4x x 3 + 4x 2 − 12 x size 12{ { {3} over {x rSup { size 8{2} } +"12"x+"36"} } = { {4x} over {x rSup { size 8{3} } +4x rSup { size 8{2} } - "12"x} } } {} Same problem we worked before, but now we are equating these two fractions, instead of subtracting them.
3 ( x ) ( x − 2 ) ( x + 6 ) 2 ( x ) ( x − 2 ) = 4x ( x + 6 ) x ( x + 6 ) 2 ( x − 2 ) size 12{ { {3 \( x \) \( x - 2 \) } over { \( x+6 \) rSup { size 8{2} } \( x \) \( x - 2 \) } } = { {4x \( x+6 \) } over {x \( x+6 \) rSup { size 8{2} } \( x - 2 \) } } } {} Rewrite both fractions with the common denominator.
3 x ( x - 2 ) = 4 x ( x + 6 ) Based on the rule above—since the denominators are equal, we can now assume the numerators are equal.
3 x 2 – 6 x = 4 x 2 + 24 x Multiply it out
x 2 + 30 x = 0 What we’re dealing with, in this case, is a quadratic equation. As always, move everything to one side...
x ( x + 30 ) = 0 ...and then factor. A common mistake in this kind of problem is to divide both sides by x ; this loses one of the two solutions.
x= 0 or x= -30 Two solutions to the quadratic equation. However, in this case, x = 0 is not valid, since it was not in the domain of the original right-hand fraction. (Why?) So this problem actually has only one solution, x = – 30 .

As always, it is vital to remember what we have found here. We started with the equation 3 x 2 + 12 x + 36 = 4x x 3 + 4x 2 − 12 x size 12{ { {3} over {x rSup { size 8{2} } +"12"x+"36"} } = { {4x} over {x rSup { size 8{3} } +4x rSup { size 8{2} } - "12"x} } } {} . We have concluded now that if you plug x = – 30 into that equation, you will get a true equation (you can verify this on your calculator). For any other value, this equation will evaluate false.

To put it another way: if you graphed the functions 3 x 2 + 12 x + 36 size 12{ { {3} over {x rSup { size 8{2} } +"12"x+"36"} } } {} and 4x x 3 + 4x 2 − 12 x size 12{ { {4x} over {x rSup { size 8{3} } +4x rSup { size 8{2} } - "12"x} } } {} , the two graphs would intersect at one point only: the point when x = – 30 .

Questions & Answers

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
hello
BenJay
hi
Method
I am eliacin, I need your help in maths
Rood
how can I help
Sir
hmm can we speak here?
Amoon
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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Source:  OpenStax, Rational expressions. OpenStax CNX. Feb 28, 2011 Download for free at http://cnx.org/content/col11278/1.2
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