# Chapter 7 Lesson 2 Solving Equations with Grouping Symbols pgs

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```							          Chapter 7 Lesson 2
Solving Equations with
Grouping Symbols
pgs. 334-338
What you will learn:
Solve equations that involve grouping
symbols
Identify equations that have no solution
or an infinite number of solutions
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
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         Time       Distance
(mph)        (hours)      (miles)
Josh         2             t           2t
Maria         10            t-1        10(t-1)
What does t represent?     The time Josh travels

Why is Maria’s time shown as t-1?
She left 1 hour later than Josh
Write an equation that represents the time
when Maria catches up to Josh.      2t = 10(t-1)
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
-8     -8
Simplify/Solve: t = 5 or 1 1/4
4
Now check the previous problem.

Josh traveled 2 miles  5 hour or 2.5 miles
hour       4
Maria traveled 1 hour less than Josh. She
traveled: 10 miles  1 hour
hour       4       or 2.5 miles
Therefore, Maria caught up to Josh in 1/4
hour or 15 minutes.
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 - 18 +18 = 8.4 + 18
Simplify: 2n = 26.4
Simplify/Solve:
Divide both sides by 2: 2n = 26.4         n = 13.2
2     2
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
• 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           Now find the area
14 + 6 = 20               Of the rectangle.
20 = 20 
A = LW
A = 73
A = 21 ft2
Example 3: No Solution

Solve: 12 - h = -h + 3
Add an h to both sides: 12 - h + h = -h + h + 3
Simplify: 12 = 3

The sentence 12 = 3 is never true. So the
Solution set is 
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 + 12 - 12 = 6g + 12 - 12
Simplify: 6g = 6g

Mentally divide each side by 6: g = g
The sentence g = g is always true, the solution set
is all numbers.
Solve each equation. Check your
solution
a = 11Check: 3(11-5) = 18
A. 3(a-5) = 18                         3(6) = 18
18 = 18 
S = 18 Check: 3(18+22) = 4(18+12)
B. 3(s+22) = 4(s+12)                 3(40) = 4(30)
120 = 120
C. 4(f+3) + 5 = 17 + 4f
f=f   The solution set is all numbers
D. 8y - 3 = 5(y - 1) +3y
The solution set is 
One More!

Find the dimension of the rectangle.
P = 460ft                2(w) + 2(w+30)=460
2w + 2w + 60 = 460
4w + 60 = 460
w 4w + 60-60 = 460 -60
4w = 400
w =100
w + 30
w + 30 =Length
100 + 30 = Length
130 = L
So the the rectangle is 130ft by 100ft
• PRACTICE IS BY THE DOOR ON

• QUIZ TOMORROW OVER 7-1 & 7-2

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