How to check if a value is a solution to an inequality Grade 7

📚 Understanding Inequalities and Solutions

An inequality is a mathematical statement that compares two expressions using symbols like < (less than), > (greater than), $\le$ (less than or equal to), or $\ge$ (greater than or equal to). A solution to an inequality is a value that, when substituted for the variable, makes the inequality true. Let's explore how to check if a given value is a solution!

📜 History of Inequalities

While the concept of inequalities has likely existed for centuries, formal notation and systematic study developed alongside algebra. Mathematicians like Thomas Harriot, who introduced symbols like '<' and '>', were instrumental in shaping how we express and solve inequalities today.

🔑 Key Principles

• 🧮 Substitution: Replace the variable in the inequality with the given value.
• ✔️ Simplification: Simplify both sides of the inequality after substitution.
• ✅ Verification: Check if the resulting inequality statement is true. If it is, the value is a solution; if not, it isn't.

📝 Step-by-Step Guide to Checking Solutions

1. 🔢 Step 1: Identify the inequality and the potential solution.
2. 🔄 Step 2: Substitute the value into the inequality.
3. ➗ Step 3: Simplify both sides of the inequality.
4. ⚖️ Step 4: Determine if the resulting inequality is true.

➕ Example 1: Is x = 3 a solution to x + 2 < 7?

1. ➡️ Substitute: $3 + 2 < 7$
2. ⬇️ Simplify: $5 < 7$
3. ✅ Verify: $5 < 7$ is true. Therefore, $x = 3$ is a solution.

➖ Example 2: Is y = 6 a solution to 2y > 10?

1. ➡️ Substitute: $2(6) > 10$
2. ⬇️ Simplify: $12 > 10$
3. ✅ Verify: $12 > 10$ is true. Therefore, $y = 6$ is a solution.

➗ Example 3: Is z = 1 a solution to 3z + 1 $\ge$ 5?

1. ➡️ Substitute: $3(1) + 1 \ge 5$
2. ⬇️ Simplify: $3 + 1 \ge 5$ becomes $4 \ge 5$
3. ❌ Verify: $4 \ge 5$ is false. Therefore, $z = 1$ is not a solution.

💡 Tips for Success

• 🧐 Pay Attention to Symbols: Make sure you understand the meaning of each inequality symbol (<, >, $\le$, $\ge$).
• ✍️ Show Your Work: Writing out each step helps prevent errors.
• ✔️ Double-Check: Always double-check your arithmetic!

📝 Practice Quiz

Determine whether the given value is a solution to the inequality:

1. Question 1: Is $x = 4$ a solution to $x + 3 < 9$?
2. Question 2: Is $y = 2$ a solution to $5y > 8$?
3. Question 3: Is $z = 7$ a solution to $z - 2 \le 5$?
4. Question 4: Is $a = 0$ a solution to $2a + 4 \ge 4$?
5. Question 5: Is $b = 10$ a solution to $\frac{b}{2} < 4$?
6. Question 6: Is $c = 3$ a solution to $4c - 1 > 10$?
7. Question 7: Is $d = 5$ a solution to $6 - d \le 2$?

✔️ Solutions to Practice Quiz

1. Answer 1: Yes, $4 + 3 < 9$ simplifies to $7 < 9$, which is true.
2. Answer 2: Yes, $5(2) > 8$ simplifies to $10 > 8$, which is true.
3. Answer 3: Yes, $7 - 2 \le 5$ simplifies to $5 \le 5$, which is true.
4. Answer 4: Yes, $2(0) + 4 \ge 4$ simplifies to $4 \ge 4$, which is true.
5. Answer 5: No, $\frac{10}{2} < 4$ simplifies to $5 < 4$, which is false.
6. Answer 6: Yes, $4(3) - 1 > 10$ simplifies to $11 > 10$, which is true.
7. Answer 7: No, $6 - 5 \le 2$ simplifies to $1 \le 2$, which is true.

⭐ Conclusion

Checking if a value is a solution to an inequality is a fundamental skill in algebra. By substituting the value, simplifying, and verifying, you can determine whether it satisfies the inequality. Keep practicing, and you'll master it in no time!