Solution: Subtract 1: $ \frac{2x - 5}{x + 3} - 1 \geq 0 \Rightarrow \frac{2x - 5 - (x + 3)}{x + 3} \geq 0 \Rightarrow \frac{x - 8}{x + 3} \geq 0 $. Critical points at $ x = 8 $ and $ x = -3 $. Test intervals:

["Solving the Inequality: $ \frac{2x - 5}{x + 3} - 1 \geq 0 $", "When solving rational inequalities, understanding how to manipulate and analyze the expression is key to finding the correct solution set. This article walks you through solving the inequality:", "$$\n\frac{2x - 5}{x + 3} - 1 \geq 0\n$$", "---", "### Step 1: Combine the Expression", "Start by combining the terms into a single rational expression. Subtract 1 from the fraction:", "$$\n\frac{2x - 5}{x + 3} - 1 = \frac{2x - 5 - (x + 3)}{x + 3} = \frac{2x - 5 - x - 3}{x + 3} = \frac{x - 8}{x + 3}\n$$", "So the inequality becomes:", "$$\n\frac{x - 8}{x + 3} \geq 0\n$$", "---", "### Step 2: Identify Critical Points", "The expression $ \frac{x - 8}{x + 3} $ is a rational function with two important critical points:", "- Numerator zero (where expression equals zero): $ x = 8 $\n- Denominator zero (where function is undefined): $ x = -3 $", "These points divide the number line into three intervals:", "1. $ (-\infty, -3) $\n2. $ (-3, 8) $\n3. $ (8, \infty) $", "Note: $ x = -3 $ is excluded from the solution because it makes the denominator zero (undefined).", "---", "### Step 3: Analyze the Sign of the Expression in Each Interval", "We now test the sign of $ \frac{x - 8}{x + 3} $ in each interval:", "| Interval | Test Point | $ x - 8 $ | $ x + 3 $ | $ \frac{x - 8}{x + 3} $ |\n|---------|------------|------------|------------|---------------------------|\n| $ (-\infty, -3) $ | $ x = -4 $ | $ -12 $ (negative) | $ -1 $ (negative) | $ \frac{-}{-} = + $ (positive) |\n| $ (-3, 8) $ | $ x = 0 $ | $ -8 $ (negative) | $ +3 $ (positive) | $ \frac{-}{+} = - $ (negative) |\n| $ (8, \infty) $ | $ x = 9 $ | $ +1 $ (positive) | $ +12 $ (positive) | $ \frac{+}{+} = + $ (positive) |", "---", "### Step 4: Determine Solution Based on Inequality", "We are solving:", "$$\n\frac{x - 8}{x + 3} \geq 0\n$$", "This inequality is satisfied when the expression is positive or zero. From the sign chart:", "- Positive in $ (-\infty, -3) $ and $ (8, \infty) $\n- Zero at $ x = 8 $ (included because inequality includes equality)\n- Undefined at $ x = -3 $ (excluded)", "---", "### Final Answer:", "$$\n\boxed{(-\infty, -3) \cup [8, \infty)}\n$$", "This is the solution set for the inequality $ \frac{2x - 5}{x + 3} - 1 \geq 0 $. Remember to exclude $ x = -3 $ due to division by zero and include $ x = 8 $ since the inequality allows equality.", "---", "Key Takeaways:", "- Combine rational expressions carefully.\n- Identify critical points: where numerator is zero or denominator is zero.\n- Use test intervals to determine sign composition.\n- Include zeros but exclude undefined points.", "Understanding these steps will help you solve similar rational inequality problems with confidence!"]