Solve the inequality 2x + 3 > 5x + 8

This problem is about solving a linear inequality. The goal is to find all values of \( x \) that make the inequality 2x + 3 > 5x + 8 true.

Let's solve it step by step:

• Start with the given inequality:
  
  \( 2x + 3 > 5x + 8 \)
• Get all the terms with \( x \) on one side and constants on the other.
  
  Subtract \( 2x \) from both sides:
  \( 2x + 3 - 2x > 5x + 8 - 2x \)
  \( 3 > 3x + 8 \)
• Subtract 8 from both sides to get constants on one side:
  
  \( 3 - 8 > 3x + 8 - 8 \)
  \( -5 > 3x \)
• Divide both sides by 3 to solve for \( x \):
  
  • Remember, when dividing or multiplying both sides of an inequality by a negative number, you must flip the inequality symbol. But we are dividing by a positive number (3), so the direction stays the same.
  \[ \frac{-5}{3} > x \] or rewritten, \[ x < -\frac{5}{3} \] As a mixed number, \( -\frac{5}{3} = -1\frac{2}{3} \).

Conclusion: The solution is \( x < -1\frac{2}{3} \). This means any value of \( x \) that is less than \( -1\frac{2}{3} \) makes the original inequality true.

Why is this true? The process of solving the inequality is just like solving equations: you isolate \( x \) using addition, subtraction, and division. The key thing in inequalities is to only flip the symbol if you multiply or divide by a negative number. Here, all steps kept the direction of the inequality the same.