Solved examples: Two-step inequalities from word problems

📚 Quick Study Guide

• 🔢 Isolating the Variable: The goal is to get the variable by itself on one side of the inequality. Use inverse operations to isolate the variable.
• ⚖️ Inequality Properties: Remember to perform the same operation on both sides of the inequality to maintain balance.
• 🔀 Reversing the Inequality: If you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign.
• ✍️ Translating Words to Math:
  • "More than" means $>$.
  • "Less than" means $<$.
  • "At least" means $\geq$.
  • "At most" means $\leq$.
• 💡 Two-Step Inequality Format: These problems can be expressed as $ax + b < c$, $ax + b > c$, $ax + b \leq c$, or $ax + b \geq c$, where $a$, $b$, and $c$ are constants.

Practice Quiz

1. What inequality represents: "Three times a number plus five is less than 14"?
   1. $3x + 5 < 14$
   2. $3x + 5 > 14$
   3. $3x - 5 < 14$
   4. $3x - 5 > 14$
2. Sarah wants to save at least $200. She already has $50 and plans to save $20 each week. Which inequality represents the number of weeks, $w$, Sarah needs to save?
   1. $20w + 50 \geq 200$
   2. $20w + 50 \leq 200$
   3. $20w - 50 \geq 200$
   4. $20w - 50 \leq 200$
3. A taxi charges $2.50 plus $0.20 per mile. How many miles, $m$, can you travel if you have $10?
   1. $0.20m + 2.50 \leq 10$
   2. $0.20m + 2.50 \geq 10$
   3. $2.50m + 0.20 \leq 10$
   4. $2.50m + 0.20 \geq 10$
4. John wants to buy a new bike that costs $150. He has already saved $30 and earns $12 per hour at his job. What inequality represents the number of hours, $h$, he needs to work?
   1. $12h + 30 \geq 150$
   2. $12h + 30 \leq 150$
   3. $12h - 30 \geq 150$
   4. $12h - 30 \leq 150$
5. Solve the inequality: $2x + 3 < 7$
   1. $x < 2$
   2. $x > 2$
   3. $x < 5$
   4. $x > 5$
6. Solve the inequality: $-3x - 5 \leq 4$
   1. $x \geq -3$
   2. $x \leq -3$
   3. $x \geq 3$
   4. $x \leq 3$
7. Which inequality has the solution $x > 4$?
   1. $-2x + 3 < -5$
   2. $-2x + 3 > -5$
   3. $2x + 3 < 11$
   4. $2x + 3 > 11$