Then $ 2x - 5 < 0 $, $ x + 3 \ge 0 $, so:

["# Solving the Inequality System: $ 2x - 5 < 0 $ and $ x + 3 \ge 0 $\nStep-by-step explanation, solution techniques, and key insights", "---", "Understanding how to solve systems of inequalities like $ 2x - 5 < 0 $ and $ x + 3 \ge 0 $ is essential for mastering algebra and laying the foundation for advanced math topics. Whether you're preparing for school, solving real-world problems, or brushing up on your skills, knowing how to analyze and solve these inequalities step-by-step can make a big difference.", "## What Are These Inequalities?", "We are given a pair of inequalities:", "1. $ 2x - 5 < 0 $\n2. $ x + 3 \ge 0 $", "These are simultaneous inequalities, meaning we must find the values of $ x $ that satisfy both conditions at the same time.", "---", "## Step 1: Solve Each Inequality Individually", "### Solving $ 2x - 5 < 0 $", "Start by isolating $ x $:", "$$\n2x - 5 < 0\n$$\nAdd 5 to both sides:\n$$\n2x < 5\n$$\nDivide both sides by 2:\n$$\nx < \frac{5}{2} \quad \ ext{or} \quad x < 2.5\n$$", "---", "### Solving $ x + 3 \ge 0 $", "Isolate $ x $:", "$$\nx + 3 \ge 0\n$$\nSubtract 3 from both sides:\n$$\nx \ge -3\n$$", "---", "## Step 2: Combine the Results (Intersection of Solutions)", "We now have two conditions:\n- $ x < 2.5 $\n- $ x \ge -3 $", "Since both must be true simultaneously, the solution is the intersection of these two intervals:", "$$\n-3 \le x < 2.5\n$$", "---", "## Final Answer", "The solution set for the system of inequalities $ 2x - 5 < 0 $ and $ x + 3 \ge 0 $ is:\n$$\n\boxed{[-3,\ 2.5)}\n$$\nIn interval notation: $ x \in [-3,\ 2.5) $", "---", "## Why This Matters – Real-World Applications", "Solving these inequalities helps in various real-life scenarios, such as:\n- Budgeting: Determining allowable spending ranges within financial limits.\n- Engineering: Finding acceptable ranges for temperature, pressure, or material strengths.\n- Fitness goal-setting: Finding body weight ranges that meet health criteria.", "---", "## Summary of Key Steps", "1. Solve $ 2x - 5 < 0 $ → $ x < 2.5 $\n2. Solve $ x + 3 \ge 0 $ → $ x \ge -3 $\n3. Combine to get $ -3 \le x < 2.5 $\n4. Use interval notation for clarity", "---", "## Want to Practice More? Try Similar Problems!", "- Solve $ 3x + 2 > 8 \quad \ ext{and} \quad x - 4 \le 1 $\n- Find $ x $ such that $ x - 5 \le 0 $ and $ 2x + 1 \ge 7 $", "Mastering these systems builds problem-solving confidence and strengthens algebraic reasoning.", "---", "Keywords: solve inequalities, $ 2x - 5 < 0 $, $ x + 3 \ge 0 $, system of inequalities, algebraic solution steps, $ x < 2.5 $, $ x \ge -3 $, real-world math applications, inequality graphing, algebra tips", "---", "Take control of your math skills — understanding inequalities opens doors to complex problem solving!"]