Solve the following compound inequality. Write the solution set using interval notation and graph it.
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Ask by Daniels Ball. in the United States

Jan 23,2025

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Solve the following compound inequality. Write the solution set using interval notation and graph it. $$ \frac{1}{3}(x+2)>2 $$ or $$ 0<3-x $$

UpStudy AI · Accepted Answer

To solve the compound inequality: $$ \frac{1}{3}(x + 2) > 2 \quad \text{or} \quad 0 < 3 - x $$ we'll solve each inequality separately and then combine the solutions. ### Solving the First Inequality: $$ \frac{1}{3}(x + 2) > 2 $$ 1. **Multiply both sides by 3 to eliminate the fraction:** $$ x + 2 > 6 $$ 2. **Subtract 2 from both sides:** $$ x > 4 $$ ### Solving the Second Inequality: $$ 0 < 3 - x $$ 1. **Add $$ x $$ to both sides to isolate terms with $$ x $$:** $$ x < 3 $$ ### Combining the Solutions: The compound inequality uses the **"or"** condition, meaning either $$ x > 4 $$ **or** $$ x < 3 $$ satisfies the inequality. ### Interval Notation: $$ (-\infty, 3) \cup (4, \infty) $$ ### Graphical Representation: Here's how the solution looks on a number line: ``` <----(==========)----( )----(==========)----> -∞ 3 4 ∞ ``` - **Open circles** at 3 and 4 indicate that these points are not included in the solution. - **Shaded regions** to the left of 3 and to the right of 4 show all real numbers that satisfy either $$ x < 3 $$ or $$ x > 4 $$. --- **Final Answer:** All real numbers x with x < 3 or x > 4. In interval notation:  ( –∞, 3 ) ∪ ( 4, ∞ ) Graph: <====● ●====> 3 4 All real numbers $$ x $$ where $$ x < 3 $$ or $$ x > 4 $$. In interval notation: $$ (-\infty, 3) \cup (4, \infty) $$.

Solve the following compound inequality. Write the solution set using interval notation and graph it.
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Solve the following compound inequality. Write the solution set using interval notation and graph it. or