6. Given that \( x \) satisfies the inequalities \( 3 x+1>-10 \) and \( 2-x>4 \), determine the integer values of \( x \). 7. Given \( A=\{x:-3

Ask by Bowen Hobbs. in Mauritius

Mar 12,2025

**6. Solve the inequalities \( 3x + 1 > -10 \) and \( 2 - x > 4 \) for integer \( x \).** 1. Start with the first inequality: \[ 3x + 1 > -10 \] Subtract 1 from both sides: \[ 3x > -11 \] Divide both sides by 3: \[ x > -\frac{11}{3} \] 2. Now, consider the second inequality: \[ 2 - x > 4 \] Subtract 2 from both sides: \[ -x > 2 \] Multiply both sides by \(-1\) (remember to reverse the inequality sign): \[ x < -2 \] 3. Combine the two inequalities: \[ -\frac{11}{3} < x < -2 \] Note that \(-\frac{11}{3} \approx -3.67\). The integers that satisfy the inequality are those greater than \(-3.67\) and less than \(-2\). The only integer in this range is: \[ x = -3 \] --- **7. Given \( A = \{x : -3 < x \leq 4\} \) and \( B = \{x : 1 < x < 7\} \), determine:** **(i) \( A \cap B \)** The intersection \( A \cap B \) includes those \( x \) that satisfy both: - \( -3 < x \leq 4 \) (from \( A \)) - \( 1 < x < 7 \) (from \( B \)) The most restrictive conditions are: - For the lower bound: \( x > 1 \) (since \( 1 > -3 \)) - For the upper bound: \( x \leq 4 \) (since \( 4 < 7 \)) Thus, \[ A \cap B = \{x : 1 < x \leq 4\} \] --- **(ii) \( A \cup B \)** The union \( A \cup B \) includes any \( x \) that belongs to either \( A \) or \( B \). We have: - From \( A \): \( -3 < x \leq 4 \) - From \( B \): \( 1 < x < 7 \) Together, the union spans all \( x \) from just above \(-3\) up to just below \( 7 \). Notice: - The lower bound is determined by \( A \): \( x > -3 \) - The upper bound is determined by \( B \): \( x < 7 \) Even though \( x = 4 \) is included from \( A \), it does not affect the overall union interval. Therefore, \[ A \cup B = \{x : -3 < x < 7\} \]