Solve the inequality \( rac{2x - 1}{x + 3} > 1 \), where \( x
["# Solve the Inequality ( \dfrac{2x - 1}{x + 3} > 1 ) – Step-by-Step Guide", "Solving rational inequalities can seem challenging at first, but with a clear approach and careful attention to restrictions, you can tackle inequalities like ( \dfrac{2x - 1}{x + 3} > 1 ) with confidence. This article provides a comprehensive, step-by-step solution to this inequality while helping you understand key concepts like domain restrictions and solution intervals.", "---", "## Understanding the Inequality", "We want to solve:", "[\n\dfrac{2x - 1}{x + 3} > 1\n]", "This inequality compares a rational expression to 1. To solve such inequalities, we first move all terms to one side to form a single rational expression compared to zero.", "---", "## Step 1: Rearrange the Inequality", "Subtract 1 from both sides:", "[\n\dfrac{2x - 1}{x + 3} - 1 > 0\n]", "Now express 1 as a fraction with the same denominator:", "[\n\dfrac{2x - 1}{x + 3} - \dfrac{x + 3}{x + 3} = \dfrac{(2x - 1) - (x + 3)}{x + 3} = \dfrac{2x - 1 - x - 3}{x + 3} = \dfrac{x - 4}{x + 3}\n]", "So the inequality becomes:", "[\n\dfrac{x - 4}{x + 3} > 0\n]", "---", "## Step 2: Determine the Domain Restrictions", "The original expression and the transformed denominator ( x + 3 ) cannot be zero. So:", "[\nx + 3 <br/>\ne 0 \Rightarrow x <br/>\ne -3\n]", "This exclusion point divides the number line into intervals. We must exclude ( x = -3 ) from the solution set.", "---", "## Step 3: Solve the Simplified Inequality", "We now solve:", "[\n\dfrac{x - 4}{x + 3} > 0\n]", "This rational expression is positive when numerator and denominator have the same sign: both positive or both negative.", "Identify critical points where numerator or denominator equals zero:", "- ( x - 4 = 0 \Rightarrow x = 4 )\n- ( x + 3 = 0 \Rightarrow x = -3 ) (excluded)", "These critical points divide the number line into three intervals:", "1. ( x < -3 )\n2. ( -3 < x < 4 )\n3. ( x > 4 )", "Test the sign of ( \dfrac{x - 4}{x + 3} ) in each interval:", "| Interval | Test Value | ( x - 4 ) | ( x + 3 ) | Sign of Expression |\n|--------------------|------------|-------------|-------------|---------------------|\n| ( x < -3 ) | ( x = -4 ) | Negative | Negative | Positive (– ÷ – = +) |\n| ( -3 < x < 4 ) | ( x = 0 ) | Negative | Positive | Negative (– ÷ + = –) |\n| ( x > 4 ) | ( x = 5 ) | Positive | Positive | Positive (+ ÷ + = +) |", "---", "## Step 4: Determine Where Inequality Holds", "We seek where the expression is greater than zero:", "- Positive region: ( x < -3 )\n- Negative region: ( -3 < x < 4 )\n- Positive again: ( x > 4 )", "But exclude ( x = -3 ), so ( x = -3 ) is not part of the solution.", "Therefore, the solution is:", "[\nx < -3 \quad \ ext{or} \quad x > 4\n]", "---", "## Final Answer", "[\n\boxed{x < -3 \quad \ ext{or} \quad x > 4}\n]", "---", "## Key Takeaways", "- Always move all terms to one side to form a inequality of the form ( \dfrac{P(x)}{Q(x)} > 0 ).\n- Identify domain restrictions (where denominator is zero).\n- Use critical points to break number line into intervals.\n- Test sign changes in intervals to determine solution regions.\n- Combine intervals where the expression is positive, excluding undefined points.", "Understanding these steps helps you solve similar rational inequalities with clarity and precision. Practice with different inequalities to build confidence!"]
