Question

1. -31 + 3x = 1 + 5x
2. 9 + x = 15 + x
3. 17x + 2 = 15 + 14x + 8
4. -1 + x + 14 = x + 13
5. -2x + 17 = -30 + x + 17
6. x + 3 = 11 + x

Explanation:

Problem 7:

Step1: Isolate x terms to one side

$-31 - 1 = 5x - 3x$

Step2: Simplify both sides

$-32 = 2x$

Step3: Solve for x

$x = \frac{-32}{2} = -16$

Problem 8:

Step1: Isolate x terms to one side

$x - x = 15 - 9$

Step2: Simplify both sides

$0 = 6$

Step3: Evaluate validity

This is a false statement, so no solution exists.

Problem 9:

Step1: Combine constants on right

$17x + 2 = 14x + 23$

Step2: Isolate x terms to left

$17x - 14x = 23 - 2$

Step3: Simplify and solve for x

$3x = 21$
$x = \frac{21}{3} = 7$

Problem 10:

Step1: Combine constants on left

$x + 13 = x + 13$

Step2: Isolate x terms to one side

$x - x = 13 - 13$

Step3: Simplify both sides

$0 = 0$

Step4: Evaluate validity

This is a true statement, so all real numbers are solutions.

Problem 11:

Step1: Combine constants on right

$-2x + 17 = x - 13$

Step2: Isolate x terms to left

$-2x - x = -13 - 17$

Step3: Simplify and solve for x

$-3x = -30$
$x = \frac{-30}{-3} = 10$

Problem 12:

Step1: Isolate x terms to one side

$x - x = 11 - 3$

Step2: Simplify both sides

$0 = 8$

Step3: Evaluate validity

This is a false statement, so no solution exists.

Answer:

1. $x = -16$
2. No solution
3. $x = 7$
4. All real numbers are solutions
5. $x = 10$
6. No solution