Solved Problems: Grade 8 One-Step Inequalities with Step-by-Step Solutions.

📚 Understanding One-Step Inequalities

One-step inequalities are mathematical statements that compare two expressions using inequality symbols. Unlike equations, which show equality, inequalities show that one value is greater than, less than, greater than or equal to, or less than or equal to another value. Solving them involves isolating the variable using inverse operations, much like solving regular equations, but with a crucial rule to remember about multiplying or dividing by negative numbers.

📜 Historical Context

The concept of inequalities has been around for centuries, evolving alongside the development of algebra. Early mathematicians used inequalities to describe ranges of possible solutions, particularly in geometric and algebraic problems. The formal notation and systematic study of inequalities became more prominent in the 17th and 18th centuries, with contributions from mathematicians like Newton and Leibniz.

➗ Key Principles for Solving

• ➕ Addition Property: If $a > b$, then $a + c > b + c$. Adding the same number to both sides does not change the inequality.
• ➖ Subtraction Property: If $a > b$, then $a - c > b - c$. Subtracting the same number from both sides does not change the inequality.
• multiplication is next
• ✖️ Multiplication Property (Positive Number): If $a > b$ and $c > 0$, then $ac > bc$. Multiplying both sides by a positive number does not change the inequality.
• ➗ Division Property (Positive Number): If $a > b$ and $c > 0$, then $\frac{a}{c} > \frac{b}{c}$. Dividing both sides by a positive number does not change the inequality.
• 🔄 Multiplication Property (Negative Number): If $a > b$ and $c < 0$, then $ac < bc$. Multiplying both sides by a negative number reverses the inequality.
• ↩️ Division Property (Negative Number): If $a > b$ and $c < 0$, then $\frac{a}{c} < \frac{b}{c}$. Dividing both sides by a negative number reverses the inequality.

📝 Solved Problems

Problem 1

Solve: $x + 5 > 10$

Solution:

1. Subtract 5 from both sides: $x + 5 - 5 > 10 - 5$
2. Simplify: $x > 5$

Problem 2

Solve: $y - 3 \leq 7$

Solution:

1. Add 3 to both sides: $y - 3 + 3 \leq 7 + 3$
2. Simplify: $y \leq 10$

Problem 3

Solve: $3z < 12$

Solution:

1. Divide both sides by 3: $\frac{3z}{3} < \frac{12}{3}$
2. Simplify: $z < 4$

Problem 4

Solve: $\frac{a}{2} \geq 4$

Solution:

1. Multiply both sides by 2: $2 \cdot \frac{a}{2} \geq 2 \cdot 4$
2. Simplify: $a \geq 8$

Problem 5

Solve: $-2b > 10$

Solution:

1. Divide both sides by -2 (and reverse the inequality): $\frac{-2b}{-2} < \frac{10}{-2}$
2. Simplify: $b < -5$

Problem 6

Solve: $6 - c \leq 1$

Solution:

1. Subtract 6 from both sides: $6 - c - 6 \leq 1 - 6$
2. Simplify: $-c \leq -5$
3. Multiply both sides by -1 (and reverse the inequality): $c \geq 5$

Problem 7

Solve: $4d + 2 < 14$

Solution:

1. Subtract 2 from both sides: $4d + 2 - 2 < 14 - 2$
2. Simplify: $4d < 12$
3. Divide both sides by 4: $\frac{4d}{4} < \frac{12}{4}$
4. Simplify: $d < 3$

🌍 Real-World Examples

• 💰 Budgeting: If you want to spend no more than $50 on groceries, and you've already spent $20, the inequality $20 + x \leq 50$ can help you determine the maximum amount ($x$) you can still spend.
• 🌡️ Temperature: To keep a chemical reaction stable, the temperature must stay below 100°C. If the current temperature is 75°C, the inequality $75 + x < 100$ can help you find out how much the temperature can increase ($x$) before it exceeds the limit.
• 🏋️ Weight Limits: An elevator has a maximum weight capacity of 1500 lbs. If 10 people are already in the elevator and each weighs an average of 140 lbs, the inequality $10 \cdot 140 + x \leq 1500$ can determine the maximum additional weight ($x$) that can be added without exceeding the limit.

💡 Conclusion

One-step inequalities are a fundamental concept in algebra with wide-ranging applications. By understanding the basic principles and practicing problem-solving, you can master these inequalities and apply them to various real-world scenarios. Remember to always consider the direction of the inequality and the impact of multiplying or dividing by negative numbers. Happy solving!