Question

solve each equation. write if it is conditional, identity, or contradiction.

1. $3(x + 1) = 3x + 3$ 2. $4x - 1 = 7$ 3. $5x + 7 = 5x - 1$
2. $\frac{3x}{4} = \frac{x + 11}{5}$

solve each absolute value equation. write your answers as a {set}.

1. $|4x + 8| = 20$ 6. $|3y + 6| + 9 = 18$
2. $|x + 1| = 5$ 8. $2|w + 6| + 2 = 14$

Explanation:

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

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.

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.

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:

Identity, and the solution is all real numbers.

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