Question: Solve the inequality $ \frac{2x - 5}{x + 3} \geq 1 $ to determine the range of temperatures where a species thrives.

["# Solving the Inequality $ \frac{2x - 5}{x + 3} \geq 1 $ to Determine Thriving Temperatures", "Understanding mathematical inequalities is essential in many real-world applications—especially when analyzing environmental conditions like temperature ranges that support biological survival. In this article, we’ll solve the inequality\n$$\n\frac{2x - 5}{x + 3} \geq 1\n$$\nand uncover the range of values (temperatures, $ x $) where a species thrives based on this model.", "---", "## Understanding the Inequality", "We are solving\n$$\n\frac{2x - 5}{x + 3} \geq 1\n$$\nwhich asks for the values of $ x $ such that the fraction is greater than or equal to 1. This common form often appears in modeling thresholds, such as optimal temperature ranges for biological activity.", "---", "## Step 1: Subtract 1 from Both Sides", "To bring all terms to one side, rewrite the inequality:\n$$\n\frac{2x - 5}{x + 3} - 1 \geq 0\n$$", "Express 1 as a fraction with the same denominator:\n$$\n\frac{2x - 5}{x + 3} - \frac{x + 3}{x + 3} = \frac{(2x - 5) - (x + 3)}{x + 3} = \frac{2x - 5 - x - 3}{x + 3} = \frac{x - 8}{x + 3}\n$$", "Now our inequality becomes:\n$$\n\frac{x - 8}{x + 3} \geq 0\n$$", "---", "## Step 2: Identify Critical Points", "The inequality $ \frac{x - 8}{x + 3} \geq 0 $ changes sign at the zeros of the numerator and denominator:", "- Numerator zero: $ x - 8 = 0 $ → $ x = 8 $\n- Denominator zero: $ x + 3 = 0 $ → $ x = -3 $ (excluded from domain, as division by zero is undefined)", "These critical points divide the real number line into intervals:\n$$\n(-\infty, -3),\quad (-3, 8),\quad (8, \infty)\n$$", "---", "## Step 3: Test Intervals to Determine Sign", "We test a value in each interval to determine where the expression is non-negative.", "1. Interval $ (-\infty, -3) $: Try $ x = -4 $\n$$\n\frac{-4 - 8}{-4 + 3} = \frac{-12}{-1} = 12 \geq 0 \quad \ ext{True}\n$$", "2. Interval $ (-3, 8) $: Try $ x = 0 $\n$$\n\frac{0 - 8}{0 + 3} = \frac{-8}{3} < 0 \quad \ ext{False}\n$$", "3. Interval $ (8, \infty) $: Try $ x = 9 $\n$$\n\frac{9 - 8}{9 + 3} = \frac{1}{12} > 0 \quad \ ext{True}\n$$", "The expression is equal to zero at $ x = 8 $ (numerator zero), which satisfies $ \geq 0 $. But $ x = -3 $ is excluded because the expression is undefined.", "---", "## Step 4: Combine Results", "The inequality $ \frac{x - 8}{x + 3} \geq 0 $ holds when:\n- $ x \in (-\infty, -3) $\n- $ x = 8 $ (included)\n- $ x \in [8, \infty) $", "So the solution set is:\n$$\n(-\infty, -3) \cup [8, \infty)\n$$", "---", "## Step 5: Interpret the Result Ecologically", "If $ x $ represents temperature (in degrees Celsius) and the inequality models environmental conditions under which a species thrives, the solution indicates:", "- Temperatures below 3°C (i.e., $ x < -3 $) are too cold; the species cannot survive.\n- Temperatures from 8°C onward support the species, particularly approaching and exceeding 8°C where metabolic processes are optimal.", "The threshold at $ x = 8 $ marks the onset of favorable conditions. Though the inequality excludes $ x = -3 $, real-world temperature modeling often treats discontinuities cautiously—limiting suitability below numerical thresholds like $ -3^\circ \mathrm{C} $ to avoid extreme cold stress.", "---", "## Conclusion", "Solving $ \frac{2x - 5}{x + 3} \geq 1 $ yields a critical temperature range for species survival:\n$$\nx \in (-\infty, -3) \cup [8, \infty)\n$$", "This indicates the species thrives in colder climates below $ 3^\circ \mathrm{C} $ and fully above $ 8^\circ \mathrm{C} $, narrowing ecological optimization to warmer thresholds. Understanding such inequality solutions supports environmental modeling, conservation planning, and biological research informed by mathematical reasoning.", "---", "🔍 Key Takeaway: When analyzing environmental inequalities like this, careful domain analysis and sign testing reveal real-world thresholds—critical for ecology, engineering, and climate science.", "---", "*Keywords: solve inequality $ \frac{2x - 5}{x + 3} \geq 1 $, temperature range for species, environmental modeling, calculus for biology, critical temperature thresholds."]