7.1 Writing and Solving One-Step Inequalities: Answers and In-Depth Explanations
Solving one-step inequalities is a fundamental concept in algebra, crucial for understanding more complex mathematical problems. This guide provides answers and detailed explanations to common problems found in Section 7.1 of many algebra textbooks, focusing on writing and solving these inequalities. We'll cover various inequality symbols, different solution methods, and how to represent solutions graphically and using interval notation. Remember that this is a supplemental resource and shouldn't replace your textbook or class notes.
Understanding Inequality Symbols
Before diving into solving inequalities, it's crucial to understand the symbols used:
- <: Less than
- >: Greater than
- ≤: Less than or equal to
- ≥: Greater than or equal to
These symbols dictate the relationship between two expressions. Remember that the "open" side of the symbol always faces the larger value.
Solving One-Step Inequalities: Methods and Examples
Solving one-step inequalities involves isolating the variable using inverse operations, similar to solving equations. However, there's one crucial difference: when multiplying or dividing by a negative number, you must reverse the inequality symbol.
Example 1: Addition/Subtraction
Problem: x + 5 < 12
Solution:
- Subtract 5 from both sides: x + 5 - 5 < 12 - 5
- Simplify: x < 7
The solution is all values of x less than 7. Graphically, this is represented by an open circle at 7 on the number line, with the line extending to the left. In interval notation, this is written as (-∞, 7).
Example 2: Multiplication/Division (Positive Number)
Problem: 3x ≥ 18
Solution:
- Divide both sides by 3: (3x)/3 ≥ 18/3
- Simplify: x ≥ 6
The solution includes all values of x greater than or equal to 6. Graphically, this is shown with a closed circle at 6 on the number line, extending to the right. In interval notation: [6, ∞).
Example 3: Multiplication/Division (Negative Number)
Problem: -2x < 10
Solution:
- Divide both sides by -2: (-2x)/-2 > 10/-2 (Notice the inequality symbol reversed!)
- Simplify: x > -5
The solution is all values of x greater than -5. The inequality sign flips because we divided by a negative number. Graphically, this is an open circle at -5, extending to the right. Interval notation: (-5, ∞).
Common Mistakes to Avoid
- Forgetting to reverse the inequality symbol: This is the most common mistake when working with negative numbers. Always double-check your work.
- Incorrectly applying inverse operations: Make sure you're using the opposite operation to isolate the variable (addition/subtraction, multiplication/division).
- Misinterpreting the solution: Carefully examine the inequality symbol and the resulting solution to ensure you're representing it correctly on the number line and in interval notation.
Practice Problems (with Answers)
Try these problems to solidify your understanding:
- y - 7 > 3 (Answer: y > 10)
- -5a ≤ 25 (Answer: a ≥ -5)
- b/4 < -2 (Answer: b < -8)
- -6 + z ≥ 1 (Answer: z ≥ 7)
- -1/3m > 6 (Answer: m < -18)
This guide provides a solid foundation for understanding and solving one-step inequalities. Remember to practice regularly and consult your textbook or teacher for further clarification. Mastering this concept is key to success in more advanced algebra topics.
