Graph the solution to the inequality [tex]n-3 \leq 7[/tex].
Answer (1)
Add 3 to both sides of the inequality: n − 3 ≤ 7 becomes n ≤ 10 .
The solution includes all numbers less than or equal to 10.
Represent 10 on a number line with a closed circle (solid dot).
Shade the region to the left of 10 to represent all numbers less than 10: n ≤ 10 .
Explanation
Understanding the Problem We are given the inequality n − 3 ≤ 7 . Our goal is to isolate n to find the solution set and then represent this solution graphically on a number line.
Isolating n To isolate n , we need to add 3 to both sides of the inequality: n − 3 + 3 ≤ 7 + 3
Simplifying the Inequality This simplifies to: n ≤ 10
Graphical Representation The solution to the inequality is all numbers less than or equal to 10. To represent this graphically, we draw a number line and mark the number 10. Since the inequality includes 'equal to', we use a closed circle (or a solid dot) at 10 to indicate that 10 is part of the solution. Then, we shade the region to the left of 10 to represent all numbers less than 10.
Examples
Understanding inequalities is crucial in various real-life scenarios. For instance, when budgeting, you might want to ensure your expenses ( n ) minus fixed costs (like $3 for transportation) are no more than $7 to stay within your budget. Graphing this inequality helps visualize all possible spending amounts that meet your financial constraint. Similarly, in project management, you might need to ensure that the time spent on a task ( n ) minus the initial setup time (3 hours) does not exceed 7 hours to meet a deadline. Visualizing this on a number line provides a clear understanding of the time constraints.
