solve each equation. write if it is conditional, identity, or contradiction.
1. $3(x + 1) = 3x + 3$ 2. $4x - 1 = 7$ 3. $5x + 7 = 5x - 1$
4. $\frac{3x}{4} = \frac{x + 11}{5}$

solve each absolute value equation. write your answers as a {set}.
5. $|4x + 8| = 20$ 6. $|3y + 6| + 9 = 18$
7. $|x + 1| = 5$ 8. $2|w + 6| + 2 = 14$

Question

solve each equation. write if it is conditional, identity, or contradiction.
1. $3(x + 1) = 3x + 3$  2. $4x - 1 = 7$  3. $5x + 7 = 5x - 1$
4. $\frac{3x}{4} = \frac{x + 11}{5}$

solve each absolute value equation. write your answers as a {set}.
5. $|4x + 8| = 20$  6. $|3y + 6| + 9 = 18$
7. $|x + 1| = 5$  8. $2|w + 6| + 2 = 14$

Accepted Answer

### Problem 1: $ 3(x + 1) = 3x + 3 $
# Explanation:
## Step 1: Expand the left side
Using the distributive property $ a(b + c) = ab + ac $, we get $ 3x + 3 = 3x + 3 $.
## Step 2: Analyze the equation
Subtract $ 3x $ from both sides: $ 3 = 3 $. This is always true, so it's an identity.
# Answer:
Identity, and the solution is all real numbers.

### Problem 2: $ 4x - 1 = 7 $
# Explanation:
## Step 1: Add 1 to both sides
$ 4x - 1 + 1 = 7 + 1 $, which simplifies to $ 4x = 8 $.
## Step 2: Divide by 4
$ \frac{4x}{4} = \frac{8}{4} $, so $ x = 2 $. Since there's one solution, it's conditional.
# Answer:
Conditional, $ x = 2 $.

### Problem 3: $ 5x + 7 = 5x - 1 $
# Explanation:
## Step 1: Subtract $ 5x $ from both sides
$ 5x + 7 - 5x = 5x - 1 - 5x $, which gives $ 7 = -1 $. This is never true, so it's a contradiction.
# Answer:
Contradiction, no solution.

### Problem 4: $ \frac{3x}{4} = \frac{x + 11}{5} $
# Explanation:
## Step 1: Cross - multiply
$ 5\times3x = 4\times(x + 11) $, so $ 15x = 4x + 44 $.
## Step 2: Subtract $ 4x $
$ 15x - 4x = 4x + 44 - 4x $, which simplifies to $ 11x = 44 $.
## Step 3: Divide by 11
$ \frac{11x}{11} = \frac{44}{11} $, so $ x = 4 $. It's conditional.
# Answer:
Conditional, $ x = 4 $.

### Problem 5: $ |4x + 8| = 20 $
# Explanation:
## Step 1: Set up two equations
Case 1: $ 4x + 8 = 20 $
Case 2: $ 4x + 8=-20 $
## Step 2: Solve Case 1
Subtract 8: $ 4x = 20 - 8 = 12 $, divide by 4: $ x = 3 $.
## Step 3: Solve Case 2
Subtract 8: $ 4x=-20 - 8=-28 $, divide by 4: $ x=-7 $.
# Answer:
$ \{ - 7,3\} $

### Problem 6: $ |3y + 6| + 9 = 18 $
# Explanation:
## Step 1: Subtract 9 from both sides
$ |3y + 6|=18 - 9 = 9 $.
## Step 2: Set up two equations
Case 1: $ 3y + 6 = 9 $
Case 2: $ 3y + 6=-9 $
## Step 3: Solve Case 1
Subtract 6: $ 3y = 9 - 6 = 3 $, divide by 3: $ y = 1 $.
## Step 4: Solve Case 2
Subtract 6: $ 3y=-9 - 6=-15 $, divide by 3: $ y=-5 $.
# Answer:
$ \{ - 5,1\} $

### Problem 7: $ |x + 1| = 5 $
# Explanation:
## Step 1: Set up two equations
Case 1: $ x + 1 = 5 $
Case 2: $ x + 1=-5 $
## Step 2: Solve Case 1
Subtract 1: $ x = 5 - 1 = 4 $.
## Step 3: Solve Case 2
Subtract 1: $ x=-5 - 1=-6 $.
# Answer:
$ \{ - 6,4\} $

### Problem 8: $ 2|w + 6| + 2 = 14 $
# Explanation:
## Step 1: Subtract 2 from both sides
$ 2|w + 6|=14 - 2 = 12 $.
## Step 2: Divide by 2
$ |w + 6| = 6 $.
## Step 3: Set up two equations
Case 1: $ w + 6 = 6 $
Case 2: $ w + 6=-6 $
## Step 4: Solve Case 1
Subtract 6: $ w = 6 - 6 = 0 $.
## Step 5: Solve Case 2
Subtract 6: $ w=-6 - 6=-12 $.
# Answer:
$ \{ - 12,0\} $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: \( 3(x + 1) = 3x + 3 \)

Step 1: Expand the left side

Step 2: Analyze the equation

Step 1: Add 1 to both sides

Step 2: Divide by 4

Step 1: Subtract \( 5x \) from both sides

Answer:

Problem 2: \( 4x - 1 = 7 \)