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Core Connections Algebra 2, 2013

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Core Connections Algebra 2, 2013 View details

1. Section 4.1

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Exercise 53 Page 184

Practice makes perfect

a The equation can be solved if we distribute 3 to each term in the parentheses and then use inverse operations to isolate x.

5-3(1/2x+2)=-7

MoveRightFacToNumOne

1/b* a = a/b

5-3(x/2+2)=-7

Distribute -3

5-3(x/2)-6=-7

MoveLeftFacToNum

a*b/c= a* b/c

5-3x/2-6=-7

LHS+3x/2=RHS+3x/2

5-6=-7+3x/2

LHS+7=RHS+7

6=3x/2

LHS * 2=RHS* 2

12=3x

Rearrange equation

3x=12

.LHS /3.=.RHS /3.

x=4

b The equation is a radical equation. It can be solved by inverse operations. Notice though that to solve for x, we have to square both sides which can introduce extraneous solutions.

5(sqrt(x-2)+1)=15

.LHS /5.=.RHS /5.

sqrt(x-2)+1=3

LHS-1=RHS-1

sqrt(x-2)=2

LHS^2=RHS^2

x-2=4

LHS+2=RHS+2

x=6

The solution to the equation is x=6. However, we need to check if it's an extraneous solution. To do this, we substitute x with 6 in the equation and see if the equality holds.

5(sqrt(x-2)+1)? =15

x= 6

5(sqrt(6-2)+1)? =15

▼

Evaluate left-hand side

Subtract term

5(sqrt(4)+1)? =15

Calculate root

5(2+1)? =15

Add terms

5(3)? =15

Multiply

15=15 ✓

Since the equality is true, x=6 is a solution to the equation.

c The first step is to remove the parentheses. Since there is a minus sign in front of it, we have to change the signs inside the parentheses when removing it. We can then solve the equation with inverse operations.

12-(2x/3+x)=2

Distribute -1

12-2x/3-x=2

a = 3* a/3

12-2x/3-3x/3=2

MoveNegFracToNum

Put minus sign in numerator

12+- 2x/3-3x/3=2

Subtract fractions

12+-5x/3=2

LHS-12=RHS-12

-5x/3=-10

LHS * 3=RHS* 3

-5x=-30

.LHS /(-5).=.RHS /(-5).

x=6

d The equation is cubic, and it can be solved by taking the cube root of both sides. However, we must first isolate the cubed expression.

-3(2x+1)^3=-192

LHS * (-1)=RHS* (-1)

3(2x+1)^3=192

.LHS /3.=.RHS /3.

(2x+1)^3=64

sqrt(LHS)=sqrt(RHS)

2x+1=4

LHS-1=RHS-1

2x=3

.LHS /2.=.RHS /2.

x=3/2

Subchapter links

4.1.1 Strategies for Solving Equations p.171-172

4.1.2 Solving Equations and Systems Graphically p.176-178

4.1.3 Finding Multiple Solutions to Systems of Equations p.180-181

4.1.4 Using Systems of Equations to Solve Problems p.184