1. Solve equations that involve grouping symbols
Chapter 7 Lesson 2 Solving Equations with Grouping Symbols pgs
What you will learn:
Solve equations that involve grouping symbols
Identify equations that have no solution or an infinite number of solutions

2. Vocabulary
Null/empty set (336): equations that have no solution. No value of the variable results in a true sentence. Represented by  or { }
Identity (336): an equation that is true for every value of the variable

3. Josh starts walking at a rate of 2 mph
Josh starts walking at a rate of 2 mph. One hour later, his sister Maria starts on the same path on her bike, riding at 10mph
Rate
(mph)
Time
(hours)
Distance
(miles)
Josh 2 t 2t
Maria 10
t-1
10(t-1)
What does t represent?
Why is Maria’s time shown as t-1?
Write an equation that represents the time when Maria catches up to Josh.
The time Josh travels
She left 1 hour later than Josh
2t = 10(t-1)

4. Example 1: Solve Equations with Parentheses
Solve the equation from the previous chart: 2t = 10(t-1)
Write the problem: 2t = 10(t -1)
Distributive Property: 2t = 10(t) - 10(1)
Simplify: 2t = 10t -10
Subtract 10t from each side: 2t -10t = 10t - 10t -10
Simplify: -8t = -10
Divide each side by -8: -8t = -10
Simplify/Solve: t = 5 or 1 1/4 4

5. Now check the previous problem.
Josh traveled 2 miles  5 hour or 2.5 miles
hour
Maria traveled 1 hour less than Josh. She traveled: 10 miles  1 hour
hour or 2.5 miles
Therefore, Maria caught up to Josh in 1/4 hour or 15 minutes.

6. Another Example 1: Solve Equations with Parentheses
Solve: 6(n-3) = 4(n + 2.1)
Distributive Property on both sides: 6(n) - 6(3) = 4(n) + 4(2.1)
Simplify: 6n - 18 = 4n + 8.4
Subtract 4n from each side: 6n -4n -18 = 4n -4n + 8.4
Simplify: 2n - 18 = 8.4
Add 18 to both sides: 2n =
Simplify: 2n = 26.4
Simplify/Solve:
n = 13.2
Divide both sides by 2: 2n = 26.4
Check your solution!

7. Example 2: Use an Equation to Solve a Problem
The perimeter of a rectangle is 20 feet. The width is 4 feet less than the length. Find the dimensions of the rectangle. Then find its area.
Words: The width is 4 feet less than the length. The perimeter
is 20 feet.
Symbols: Let A = area
Let L-4 = width
2length + 2width = perimeter
Equation: 2length + 2(L-4) = 20
2L + 2L - 8 = 20
4L - 8 = 20
4L = 28
L = 7

8. Since we know the length is 7 ft, now we need to find the width.
Formula: 2L + 2W = Perimeter
2(7) + 2W = 20
14 + 2W = 20
2W = 6
W = 3 So the width is 3 feet
Check: 2(7) + 2(3) = 20
= 20
20 = 20 
Now find the area
Of the rectangle.
A = LW
A = 73
A = 21 ft2

9. Example 3: No Solution Solve: 12 - h = -h + 3
Add an h to both sides: h + h = -h + h + 3
Simplify: 12 = 3
The sentence 12 = 3 is never true. So the
Solution set is 

10. Example 4: All Numbers as solutions Remember, an equation that is true for every value of the variable is called an identity.
Solve: 3(2g + 4) = 6(g+2)
Distributive Property: 3(2g) + 3(4) = 6(g) + 6(2)
Simplify: 6g + 12 = 6g + 12
Subtract 12 from each side: 6g = 6g
Simplify: 6g = 6g
Mentally divide each side by 6: g = g
The sentence g = g is always true, the solution set
is all numbers.

11. Your Turn! Solve each equation. Check your solution
a = Check: 3(11-5) = 18
3(6) = 18
18 = 18 
3(a-5) = 18
3(s+22) = 4(s+12)
4(f+3) + 5 = f
8y - 3 = 5(y - 1) +3y
S = 18 Check: 3(18+22) = 4(18+12)
3(40) = 4(30)
120 = 120
f = f The solution set is all numbers
The solution set is 

12. One More! Find the dimension of the rectangle. P = 460ft
2(w) + 2(w+30)=460
2w + 2w + 60 = 460
4w + 60 = 460
4w =
4w = 400
w =100 w
w + 30
w + 30 =Length
= Length
130 = L
So the the rectangle is 130ft by 100ft

13. PRACTICE IS BY THE DOOR ON YOUR WAY OUT!
QUIZ TOMORROW OVER 7-1 & 7-2