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- 1 - Simplifying Expressions Using the Distributive Property—Explanation & Practice 1/7/15—mm-fd
Simplifying Expressions Using the Distributive Property—
Explanation & Practice
The problem 5 ( 3 + 4 ) can be simplified in two ways.
Method 1: First find the sum inside the parentheses. 5 ( 3 + 4 ) Then multiply the sum by 5. 5 ( 7 ) 35
Method 2: First multiply each number inside the parentheses by 5 5 ( 3 + 4 ) ( 5 ) ( 3 ) + ( 5 ) ( 4 ) Find the sum of the two products. 15 + 20 35
In both cases, the answer is 35. Example 1. Simplify –4 ( x – 3). The Order of Operations tells us to simplify, if possible, inside the grouping symbols. However, inside the parenthesis x and –3 are not like terms, so they cannot be combined. Next, we use the Distributive Property to multiply the quantity ( x – 3) by (–4). Each term inside the 1st^ parenthesis is multiplied by (–4). Arrows may help you remember this:
- 4 ( x – 3 ) Notice that the arrows go from sign to sign to aid in using the sign rules for multiplication accurately. After multiplying, the parentheses are removed.
–4 ( x – 3)
= (–4)( x ) and (–4)( –3) This step may be done mentally.
= –4 x + 12
Example 2. Simplify 2(3 x – 4) –7( x + 1).
2(3 x – 4) –7( x + 1). Nothing can be simplified inside the grouping symbols.
2(3 x – 4) –7( x + 1) Multiply (3 x – 4) by 2 and ( x + 1) by –7. Note : multiply by negative seven, not seven.
2(3 x ) and 2(–4) and –7( x ) and –7(+1) Think this step.
6 x – 8 – 7 x – 7 Remove the parentheses. Now add like terms.
6 x + (–7 x ) and –8 + (–7) Think this step.
- 2 - Simplifying Expressions Using the Distributive Property—Explanation & Practice 1/7/15—mm-fd
Example 3. Simplify 2[4 – 3(7 x + 5)].
Here we start with the embedded grouping symbols. Inside the parentheses, nothing can be simplified. The first operation we are able to perform is to multiply the quantity (7 x + 5) by –3. Again, arrows may help your accuracy.
2[4 – 3(7 x + 5)] = 2[4 – 21 x – 15] The parentheses are removed upon doing the multiplication.
Now we have only one grouping symbol: brackets. Using the Order of Operations, simplify the expression inside the brackets before going outside the brackets. The expression [4 – 21 x – 15] can be simplified by adding like terms, so: 2[4 – 21 x –15] = 2[–21 x – 11]
Having simplified inside the brackets, we next multiply [–21 x – 11] by 2. 2[–21 x – 11] = –42 x – 22 Simplified! No addition can be done.
Example 4. Simplify –3 [2 x – 4( x + 2) – 3].
–3 [2 x – 4( x + 2) – 3] Nothing can be simplified inside the parentheses. Multiply ( x + 2) by -4 and remove the parentheses but copy everything else.
= –3[2 x – 4 x – 8 – 3] Simplify the expression inside the brackets. The only possible operation inside the brackets is to add like terms.
= –3[-2 x – 11] Now go outside the brackets. Multiply and remove the brackets.
= 6 x + 33 Simplified!
Practice Exercises. Simplify.
- 2 ( x + 5)
- 3 ( 6 x – 4 )
- –5 ( 2 x + 1 )
- –3 ( –4 x – 7 + 5 ) [Careful! Simplify inside first.]
- 2 ( 3 x + 4 ) – 5
- 8 + 4 ( –2 x + 3 )
- 7 ( 2 x – 3 ) + 5 ( x + 4 )
- 3 ( x – 4 ) – 2 ( 2 x + 1 )
- –4 ( 2 x + 5 ) – 3 ( – x + 6 )
- 2 [ 8 x + 2 ( 3 x – 1 ) ]
- –5 [ 10 – 4 ( 2 x + 3 ) ]
- –4 [ 9 x – 3 ( 2 x – 1 ) – 8 ]
4 Simplifying Expressions Using the Distributive Property—Explanation & Practice 1/7/15—mm-fd
Review
■ Simplify
1. 5 x – 3 ( 2 x +7 ) 2. 3 a – ( 4 a + 1 ) 3. 8 – 5 ( 2 y +3 ) 4. 9 – ( 10 x – 4 ) 5. 7 – 2 ( 3 x - 5 ) 6. 6 – ( 8 +5 y ) 7. 4n – ( 6 – 3n ) 8. 3 x – ( 10 – x ) 9. 2 ( x + 1 ) – 4 ( x – 6 ) 10. 3 ( x – 3 ) – 2 ( x + 4 ) 11. 8 ( y – 1 ) + 2 ( 5 – 2 y ) 12. 5 (2 y – 5 ) – 2 ( 4 – y ) 13. 2 ( x + y ) – 3( x – y ) 14. 3 ( a + b ) – ( a – 2 b ) 15. 3 [ x – 3 ( x – 2 ) ] 16. 5 [ x + 3 ( x + 6 ) ] 17. – 3 [ 2 x + 3 ( 2 – x ) ] 18. –4 [ x + 2 ( 6 – x ) ] 19. –2 [ 3 x – ( x + 8 ) ] 20. –5 [ 2 x – ( 4 x – 1 ) ] 21. 3 x – 2 [ x – 3 ( 5 – x ) ] 22. –6 x +4 [ x – 6 (2 – x ) ]
23 –4 x – 3 [ 3 x – 3 ( x + 6) ] – 5 24. 3 a – 2 [ 3 b –( 2 b – a ) ] + 4 b
25. –4 x – [ 3 x – 2 ( x +6) ] – 7 26. 3 a – 3 [ b – ( 2 b – a ) ] + 5 b 27. 3 x + 2 ( x – 2 y ) + 4 ( 2 x – 5 y ) 28. 4 y – 2 ( y – 2 x ) + 3 ( 6 x – y )
A nswers
- x – 21
- a – 1
- 7 – 10 y 13 – 10 x 17 – 6 x –2 – 5 y 7n – 6 4 x – 10 –2 x + 26 x – 17 4 y + 2 12 y – 33
- x + 5 y 2 a + 5 b
- 6 x + 18 20 x + 90 3 x – 18 4 x – 48 –4 x + 16 10 x – 5 –5 x + 30 22 x – 48 –4 x + 49 a + 2 b –5 x + 5 8 b 13 x – 24 y 22 x – y
