6. Given that $$ x $$ satisfies the inequalities $$ 3 x+1-10 $$ and $$ 2-x4 $$, determine the integer values of $$ x $$. 7. Given $$ A=\{x:-3

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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

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**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, $$ A \cup B = \{x : -3 < x < 7\} $$ **6. Solve the inequalities $$ 3x + 1 > -10 $$ and $$ 2 - x > 4 $$ for integer $$ x $$.** - From $$ 3x + 1 > -10 $$: $$ x > -\frac{11}{3} \approx -3.67 $$ - From $$ 2 - x > 4 $$: $$ x < -2 $$ - Combining both: $$ -3.67 < x < -2 $$ - The only integer in this range is: $$ x = -3 $$ **7. Given $$ A = \{x : -3 < x \leq 4\} $$ and $$ B = \{x : 1 < x < 7\} $$:** **(i) $$ A \cap B $$:** $$ A \cap B = \{x : 1 < x \leq 4\} $$ **(ii) $$ A \cup B $$:** $$ A \cup B = \{x : -3 < x < 7\} $$

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

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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