<< Chapter < Page Chapter >> Page >

Simplify: 3 ( x + 4 ) .

Solution

3 ( x + 4 ) Distribute. 3 · x + 3 · 4 Multiply. 3 x + 12

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

Simplify: 4 ( x + 2 ) .

4 x + 8

Got questions? Get instant answers now!

Simplify: 6 ( x + 7 ) .

6 x + 42

Got questions? Get instant answers now!

Some students find it helpful to draw in arrows to remind them how to use the distributive property. Then the first step in [link] would look like this:

We have the expression 3 times (x plus 4) with two arrows coming from the 3. One arrow points to the x, and the other arrow points to the 4.

Simplify: 8 ( 3 8 x + 1 4 ) .

Solution

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

Simplify: 6 ( 5 6 y + 1 2 ) .

5 y + 3

Got questions? Get instant answers now!

Simplify: 12 ( 1 3 n + 3 4 ) .

4 n + 9

Got questions? Get instant answers now!

Using the distributive property as shown in [link] will be very useful when we solve money applications in later chapters.

Simplify: 100 ( 0.3 + 0.25 q ) .

Solution

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

Simplify: 100 ( 0.7 + 0.15 p ) .

70 + 15 p

Got questions? Get instant answers now!

Simplify: 100 ( 0.04 + 0.35 d ) .

4 + 35 d

Got questions? Get instant answers now!

When we distribute a negative number, we need to be extra careful to get the signs correct!

Simplify: −2 ( 4 y + 1 ) .

Solution

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

Simplify: −3 ( 6 m + 5 ) .

−18 m 15

Got questions? Get instant answers now!

Simplify: −6 ( 8 n + 11 ) .

−48 n 66

Got questions? Get instant answers now!

Simplify: −11 ( 4 3 a ) .

Solution

Distribute. .
Multiply. .
Simplify. .

Notice that you could also write the result as 33 a 44 . Do you know why?

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

Simplify: −5 ( 2 3 a ) .

−10 + 15 a

Got questions? Get instant answers now!

Simplify: −7 ( 8 15 y ) .

−56 + 105 y

Got questions? Get instant answers now!

[link] will show how to use the distributive property to find the opposite of an expression.

Simplify: ( y + 5 ) .

Solution

( y + 5 ) Multiplying by −1 results in the opposite. −1 ( y + 5 ) Distribute. −1 · y + ( −1 ) · 5 Simplify. y + ( −5 ) y 5

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

Simplify: ( z 11 ) .

z + 11

Got questions? Get instant answers now!

Simplify: ( x 4 ) .

x + 4

Got questions? Get instant answers now!

There will be times when we’ll need to use the distributive property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

Simplify: 8 2 ( x + 3 ) .

Be sure to follow the order of operations. Multiplication comes before subtraction, so we will distribute the 2 first and then subtract.

Solution

8 2 ( x + 3 ) Distribute. 8 2 · x 2 · 3 Multiply. 8 2 x 6 Combine like terms. −2 x + 2

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

Simplify: 9 3 ( x + 2 ) .

3 3 x

Got questions? Get instant answers now!

Simplify: 7 x 5 ( x + 4 ) .

2 x 20

Got questions? Get instant answers now!

Simplify: 4 ( x 8 ) ( x + 3 ) .

Solution

4 ( x 8 ) ( x + 3 ) Distribute. 4 x 32 x 3 Combine like terms. 3 x 35

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

Simplify: 6 ( x 9 ) ( x + 12 ) .

5 x 66

Got questions? Get instant answers now!

Simplify: 8 ( x 1 ) ( x + 5 ) .

7 x 13

Got questions? Get instant answers now!

All the properties of real numbers we have used in this chapter are summarized in [link] .

Commutative Property
   of addition  If a , b are real numbers, then

   of multiplication  If a , b are real numbers, then
a + b = b + a

a · b = b · a
Associative Property
   of addition  If a , b , c are real numbers, then

   of multiplication  If a , b , c are real numbers, then
( a + b ) + c = a + ( b + c )

( a · b ) · c = a · ( b · c )
Distributive Property
  If a , b , c are real numbers, then a ( b + c ) = a b + a c
Identity Property
   of addition  For any real number a :
   0 is the additive identity

   of multiplication  For any real number a :
    1 is the multiplicative identity
a + 0 = a 0 + a = a

a · 1 = a 1 · a = a
Inverse Property
   of addition  For any real number a ,
    a is the additive inverse of a

   of multiplication  For any real number a , a 0
    1 a is the multiplicative inverse of a .
a + ( a ) = 0


a · 1 a = 1
Properties of Zero
  For any real number a ,



  For any real number a , a 0

  For any real number a , a 0
a · 0 = 0 0 · a = 0

0 a = 0

a 0 is undefined

Key concepts

  • Commutative Property of
    • Addition: If a , b are real numbers, then a + b = b + a .
    • Multiplication: If a , b are real numbers, then a · b = b · a . When adding or multiplying, changing the order gives the same result.
  • Associative Property of
    • Addition: If a , b , c are real numbers, then ( a + b ) + c = a + ( b + c ) .
    • Multiplication: If a , b , c are real numbers, then ( a · b ) · c = a · ( b · c ) .
      When adding or multiplying, changing the grouping gives the same result.
  • Distributive Property: If a , b , c are real numbers, then
    • a ( b + c ) = a b + a c
    • ( b + c ) a = b a + c a
    • a ( b c ) = a b a c
    • ( b c ) a = b a c a
  • Identity Property
    • of Addition: For any real number a : a + 0 = a 0 + a = a
      0 is the additive identity
    • of Multiplication: For any real number a : a · 1 = a 1 · a = a
      1 is the multiplicative identity
  • Inverse Property
    • of Addition: For any real number a , a + ( a ) = 0 . A number and its opposite add to zero. a is the additive inverse of a .
    • of Multiplication: For any real number a , ( a 0 ) a · 1 a = 1 . A number and its reciprocal multiply to one. 1 a is the multiplicative inverse of a .
  • Properties of Zero
    • For any real number a ,
      a · 0 = 0 0 · a = 0 – The product of any real number and 0 is 0.
    • 0 a = 0 for a 0 – Zero divided by any real number except zero is zero.
    • a 0 is undefined – Division by zero is undefined.

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, 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