Question

8.3 make use of structure
for each equation, determine whether it has
or is true for all values of x (and has infinitely
has one solution, solve to find the value of x

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. $x - 3 = 3 + x$

Explanation:

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1. a.

Step1: Subtract $6x$ from both sides

$6x + 8 - 6x = 7x + 13 - 6x$
$8 = x + 13$

Step2: Subtract 13 from both sides

$8 - 13 = x + 13 - 13$
$-5 = x$

1. b.

Step1: Expand the right-hand side

$6x + 8 = 2(3x) + 2(4)$
$6x + 8 = 6x + 8$

Step2: Simplify the equation

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

1. c.

Step1: Subtract $6x$ from both sides

$6x + 8 - 6x = 6x + 13 - 6x$
$8 = 13$

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2. a.

Step1: Expand the left-hand side

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

Step2: Simplify the equation

$3 - x + x = 3 - x + x$
$3 = 3$

2. b.

Step1: Add $x$ to both sides

$x - 3 + x = 3 - x + x$
$2x - 3 = 3$

Step2: Add 3 to both sides

$2x - 3 + 3 = 3 + 3$
$2x = 6$

Step3: Divide by 2

$\frac{2x}{2} = \frac{6}{2}$
$x = 3$

2. c.

Step1: Subtract $x$ from both sides

$x - 3 - x = 3 + x - x$
$-3 = 3$

Answer:

1. a. One solution: $x = -5$
2. b. Infinitely many solutions (true for all $x$)
3. c. No solution
4. a. Infinitely many solutions (true for all $x$)
5. b. One solution: $x = 3$
6. c. No solution