Question

activity
8.3 make use of structure
for each equation, determine whether it has no solutions, exactly one solution,
or is true for all values of x (and has infinitely many solutions). if an equation
has one solution, solve to find the value of x that makes the statement true.

1. a. $6x + 8 = 7x + 13$

b. $6x + 8 = 2(3x + 4)$
c. $6x + 8 = 6x + 13$

1. a. $\frac{1}{4}(12 - 4x) = 3 - x$

b. $x - 3 = 3 - x$
c. $x - 3 = 3 + x$

1. a. $-5x - 3x + 2 = -8x + 2$

b. $-5x - 3x - 4 = -8x + 2$
c. $-5x - 4x - 2 = -8x + 2$

1. a. $4(2x - 2) + 2 = 4(x - 2)$

b. $4x + 2(2x - 3) = 8(x - 1)$
c. $4x + 2(2x - 3) = 4(2x - 2) + 2$

Explanation:

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1. a.

Step1: Isolate x terms

$6x - 7x = 13 - 8$

Step2: Simplify both sides

$-x = 5$

Step3: Solve for x

$x = -5$

1. b.

Step1: Expand right-hand side

$6x + 8 = 6x + 8$

Step2: Simplify equation

$0 = 0$

1. c.

Step1: Isolate x terms

$6x - 6x = 13 - 8$

Step2: Simplify both sides

$0 = 5$

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2. a.

Step1: Expand left-hand side

$3 - x = 3 - x$

Step2: Simplify equation

$0 = 0$

2. b.

Step1: Isolate x terms

$x + x = 3 + 3$

Step2: Simplify both sides

$2x = 6$

Step3: Solve for x

$x = 3$

2. c.

Step1: Isolate x terms

$x - x = 3 + 3$

Step2: Simplify both sides

$0 = 6$

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3. a.

Step1: Combine like terms

$-8x + 2 = -8x + 2$

Step2: Simplify equation

$0 = 0$

3. b.

Step1: Combine like terms

$-8x - 4 = -8x + 2$

Step2: Isolate x terms

$-8x + 8x = 2 + 4$

Step3: Simplify both sides

$0 = 6$

3. c.

Step1: Combine like terms

$-9x - 2 = -8x + 2$

Step2: Isolate x terms

$-9x + 8x = 2 + 2$

Step3: Simplify both sides

$-x = 4$

Step4: Solve for x

$x = -4$

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4. a.

Step1: Expand both sides

$8x - 8 + 2 = 4x - 8$

Step2: Simplify constants

$8x - 6 = 4x - 8$

Step3: Isolate x terms

$8x - 4x = -8 + 6$

Step4: Simplify both sides

$4x = -2$

Step5: Solve for x

$x = \frac{-2}{4} = -\frac{1}{2}$

4. b.

Step1: Expand both sides

$4x + 4x - 6 = 8x - 8$

Step2: Combine like terms

$8x - 6 = 8x - 8$

Step3: Isolate x terms

$8x - 8x = -8 + 6$

Step4: Simplify both sides

$0 = -2$

4. c.

Step1: Expand both sides

$4x + 4x - 6 = 8x - 8 + 2$

Step2: Combine like terms

$8x - 6 = 8x - 6$

Step3: Simplify equation

$0 = 0$

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Answer:

1. a. Exactly one solution: $\boldsymbol{x=-5}$
2. b. Infinitely many solutions (true for all $x$)
3. c. No solutions
4. a. Infinitely many solutions (true for all $x$)
5. b. Exactly one solution: $\boldsymbol{x=3}$
6. c. No solutions
7. a. Infinitely many solutions (true for all $x$)
8. b. No solutions
9. c. Exactly one solution: $\boldsymbol{x=-4}$
10. a. Exactly one solution: $\boldsymbol{x=-\frac{1}{2}}$
11. b. No solutions
12. c. Infinitely many solutions (true for all $x$)