<< Chapter < Page Chapter >> Page >

Subtract: x 2 x + 3 9 x + 3 .

x 3

Got questions? Get instant answers now!

Subtract: 4 x 2 2 x 5 25 2 x 5 .

2 x + 5

Got questions? Get instant answers now!

Be careful of the signs when you subtract a binomial!

Subtract: y 2 y 6 2 y + 24 y 6 .

Solution

y 2 y 6 2 y + 24 y 6 The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator. y 2 ( 2 y + 24 ) y 6 Distribute the sign in the numerator. y 2 2 y 24 y 6 Factor the numerator. ( y 6 ) ( y + 4 ) y 6 Remove common factors. ( y 6 ) ( y + 4 ) y 6 Simplify. y + 4

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

Subtract: n 2 n 4 n + 12 n 4 .

n + 3

Got questions? Get instant answers now!

Subtract: y 2 y 1 9 y 8 y 1 .

y 8

Got questions? Get instant answers now!

Subtract: 5 x 2 7 x + 3 x 2 3 x + 18 4 x 2 + x 9 x 2 3 x + 18 .

Solution

5 x 2 7 x + 3 x 2 3 x + 18 4 x 2 + x 9 x 2 3 x + 18 Subtract the numerators and place the difference over the common denominator. 5 x 2 7 x + 3 ( 4 x 2 + x 9 ) x 2 3 x + 18 Distribute the sign in the numerator. 5 x 2 7 x + 3 4 x 2 x + 9 x 2 3 x 18 Combine like terms. x 2 8 x + 12 x 2 3 x 18 Factor the numerator and the denominator. ( x 2 ) ( x 6 ) ( x + 3 ) ( x 6 ) Simplify by removing common factors. ( x 2 ) ( x 6 ) ( x + 3 ) ( x 6 ) Simplify. ( x 2 ) ( x + 3 )

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

Subtract: 4 x 2 11 x + 8 x 2 3 x + 2 3 x 2 + x 3 x 2 3 x + 2 .

x 11 x 2

Got questions? Get instant answers now!

Subtract: 6 x 2 x + 20 x 2 81 5 x 2 + 11 x 7 x 2 81 .

x 3 x + 9

Got questions? Get instant answers now!

Add and subtract rational expressions whose denominators are opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by −1 −1 .

Let’s see how this works.

.
Multiply the second fraction by −1 −1 . .
The denominators are the same. .
Simplify. .

Add: 4 u 1 3 u 1 + u 1 3 u .

Solution

.
The denominators are opposites, so multiply the second fraction by −1 −1 . .
Simplify the second fraction. .
The denominators are the same. Add the numerators. .
Simplify. .
Simplify. .

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

Add: 8 x 15 2 x 5 + 2 x 5 2 x .

3

Got questions? Get instant answers now!

Add: 6 y 2 + 7 y 10 4 y 7 + 2 y 2 + 2 y + 11 7 4 y .

y + 3

Got questions? Get instant answers now!

Subtract: m 2 6 m m 2 1 3 m + 2 1 m 2 .

Solution

.
The denominators are opposites, so multiply the second fraction by −1 −1 . .
Simplify the second fraction. .
The denominators are the same. Subtract the numerators. .
Distribute. m 2 6 m + 3 m + 2 m 2 1
Combine like terms. .
Factor the numerator and denominator. .
Simplify by removing common factors. .
Simplify. .

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

Subtract: y 2 5 y y 2 4 6 y 6 4 y 2 .

y + 3 y + 2

Got questions? Get instant answers now!

Subtract: 2 n 2 + 8 n 1 n 2 1 n 2 7 n 1 1 n 2 .

3 n 2 n 1

Got questions? Get instant answers now!

Key concepts

  • Rational Expression Addition
    • If p , q , and r are polynomials where r 0 , then
      p r + q r = p + q r
    • To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
  • Rational Expression Subtraction
    • If p , q , and r are polynomials where r 0 , then
      p r q r = p q r
    • To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

Practice makes perfect

Add Rational Expressions with a Common Denominator

In the following exercises, add.

3 a a b + 1 a b

3 a + 1 a + b

Got questions? Get instant answers now!

3 c 4 c 5 + 5 4 c 5

Got questions? Get instant answers now!

d d + 8 + 5 d + 8

d + 5 d + 8

Got questions? Get instant answers now!

p 2 + 10 p p + 2 + 16 p + 2

p + 8

Got questions? Get instant answers now!

q 2 + 12 q q + 3 + 27 q + 3

Got questions? Get instant answers now!

2 r 2 2 r 1 + 15 r 8 2 r 1

r + 8

Got questions? Get instant answers now!

3 s 2 3 s 2 + 13 s 10 3 s 2

Got questions? Get instant answers now!

8 t 2 t + 4 + 32 t t + 4

8 t

Got questions? Get instant answers now!

6 v 2 v + 5 + 30 v v + 5

Got questions? Get instant answers now!

2 w 2 w 2 16 + 8 w w 2 16

2 w w 4

Got questions? Get instant answers now!

7 x 2 x 2 9 + 21 x x 2 9

Got questions? Get instant answers now!

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

y 2 y + 8 64 y + 8

y 8

Got questions? Get instant answers now!

9 a 2 3 a 7 49 3 a 7

3 a + 7

Got questions? Get instant answers now!

25 b 2 5 b 6 36 5 b 6

Got questions? Get instant answers now!

c 2 c 8 6 c + 16 c 8

c + 2

Got questions? Get instant answers now!

d 2 d 9 6 d + 27 d 9

Got questions? Get instant answers now!

3 m 2 6 m 30 21 m 30 6 m 30

m 2 3

Got questions? Get instant answers now!

2 n 2 4 n 32 30 n 16 4 n 32

Got questions? Get instant answers now!

6 p 2 + 3 p + 4 p 2 + 4 p 5 5 p 2 + p + 7 p 2 + 4 p 5

p + 3 p + 5

Got questions? Get instant answers now!

5 q 2 + 3 q 9 q 2 + 6 q + 8 4 q 2 + 9 q + 7 q 2 + 6 q + 8

Got questions? Get instant answers now!

5 r 2 + 7 r 33 r 2 49 4 r 2 5 r 30 r 2 49

r + 9 r + 7

Got questions? Get instant answers now!

7 t 2 t 4 t 2 25 6 t 2 + 2 t 1 t 2 25

Got questions? Get instant answers now!

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

10 v 2 v 1 + 2 v + 4 1 2 v

4

Got questions? Get instant answers now!

20 w 5 w 2 + 5 w + 6 2 5 w

Got questions? Get instant answers now!

10 x 2 + 16 x 7 8 x 3 + 2 x 2 + 3 x 1 3 8 x

x + 2

Got questions? Get instant answers now!

6 y 2 + 2 y 11 3 y 7 + 3 y 2 3 y + 17 7 3 y

Got questions? Get instant answers now!

In the following exercises, subtract.

z 2 + 6 z z 2 25 3 z + 20 25 z 2

z + 4 z 5

Got questions? Get instant answers now!

a 2 + 3 a a 2 9 3 a 27 9 a 2

Got questions? Get instant answers now!

2 b 2 + 30 b 13 b 2 49 2 b 2 5 b 8 49 b 2

4 b 3 b 7

Got questions? Get instant answers now!

c 2 + 5 c 10 c 2 16 c 2 8 c 10 16 c 2

Got questions? Get instant answers now!

Everyday math

Sarah ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. If r represents Sarah’s speed when she ran, then her running time is modeled by the expression 8 r and her biking time is modeled by the expression 24 r + 4 . Add the rational expressions 8 r + 24 r + 4 to get an expression for the total amount of time Sarah ran and biked.

32 r + 32 r ( r + 4 )

Got questions? Get instant answers now!

If Pete can paint a wall in p hours, then in one hour he can paint 1 p of the wall. It would take Penelope 3 hours longer than Pete to paint the wall, so in one hour she can paint 1 p + 3 of the wall. Add the rational expressions 1 p + 1 p + 3 to get an expression for the part of the wall Pete and Penelope would paint in one hour if they worked together.

Got questions? Get instant answers now!

Writing exercises

Donald thinks that 3 x + 4 x is 7 2 x . Is Donald correct? Explain.

Got questions? Get instant answers now!

Explain how you find the Least Common Denominator of x 2 + 5 x + 4 and x 2 16 .

Got questions? Get instant answers now!

Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “add rational expressions with a common denominator.”, the next row reads “subtract rational expressions with a common denominator.”, the next row reads, “add and subtract rational expressions whose denominators are opposites.”, the last row reads “What does this checklist tell you about your mastery of this section? What steps will you take to improve?” The remaining columns are blank.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Questions & Answers

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
rectangular field solutions
Navin Reply
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from $1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
Kirisma Reply
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found $7(+7). Totally you lost just $12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
where's the answers?
Ed Reply
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply

Get the best Elementary algebra course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask