Question

(lessons 7-8)
how can you tell if an equation has:
a) no solutions
b) an infinite number of solutions
c) exactly one solution
taken from lesson 8 activity 3
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. $4(2x - 2) + 2 = 4(x - 2)$

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

Explanation:

1a. Step1: Isolate x terms

Subtract $6x$ from both sides:
$8 = x + 13$

1a. Step2: Solve for x

Subtract 13 from both sides:
$x = 8 - 13 = -5$

1b. Step1: Expand right-hand side

$6x + 8 = 6x + 8$

1b. Step2: Compare both sides

Identical expressions, so true for all $x$.

1c. Step1: Isolate x terms

Subtract $6x$ from both sides:
$8 = 13$

1c. Step2: Evaluate the statement

False statement, no solutions.

4a. Step1: Expand both sides

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

4a. Step2: Isolate x terms

Subtract $4x$ from both sides:
$4x - 6 = -8$

4a. Step3: Solve for x

Add 6 to both sides, then divide by 4:
$4x = -2$
$x = \frac{-2}{4} = -\frac{1}{2}$

4b. Step1: Expand both sides

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

4b. Step2: Isolate x terms

Subtract $8x$ from both sides:
$-6 = -8$

4b. Step3: Evaluate the statement

False statement, no solutions.

4c. Step1: Expand both sides

$4x + 4x - 6 = 8x - 8$
Simplify: $8x - 6 = 8x - 8$
Note: The right-hand side is cut off, but based on matching structure to 4b, the simplified form is identical to 4b

4c. Step2: Isolate x terms

Subtract $8x$ from both sides:
$-6 = -8$

4c. Step3: Evaluate the statement

False statement, no solutions.

Answer:

1. a. Exactly one solution: $x = -5$

b. Infinitely many solutions
c. No solutions

1. a. Exactly one solution: $x = -\frac{1}{2}$

b. No solutions
c. No solutions