Suppose x is an integer. Consider the proposition, “x is an even integer if and only if 5x+3 is an odd integer.” Express this proposition as a conjunction of two distinct conditional statements.

1. Identify the implications of the statement "x is an even integer if and only if 5x+3 is an odd integer." This is a biconditional statement, which means it can be expressed as two conditional statements (implications).

2. Break down the biconditional statement into its two conditional components:

- Conditional 1: If \( x \) is an even integer (hypothesis), then \( 5x+3 \) is an odd integer (conclusion).

- Conditional 2: If \( 5x+3 \) is an odd integer (hypothesis), then \( x \) is an even integer (conclusion).

3. Combine the two conditionals with "and" to form the conjunction of the conditionals:

- "If \( x \) is an even integer, then \( 5x+3 \) is an odd integer" \( \land \) "If \( 5x+3 \) is an odd integer, then \( x \) is an even integer."

**Answer:**

If \( x \) is an even integer, then \( 5x+3 \) is an odd integer.

If \( 5x+3 \) is an odd integer, then \( x \) is an even integer.