Solve This Inequality 9h 2 79

Understanding and Solving the Inequality 9h < 79

In algebra, inequalities are mathematical expressions that show a relationship where one side is greater than, less than, or not equal to the other side. The inequality 9h < 79 is a linear inequality, meaning it involves a variable raised to the first power. Solving such inequalities requires isolating the variable to determine the range of values that satisfy the condition.

Step 1: Identify the Goal

The objective is to isolate the variable h on one side of the inequality. In this case, h is being multiplied by 9, so the first step is to divide both sides by 9 to undo that multiplication.

Step 2: Perform the Division

Dividing both sides by 9 gives: [ \frac{9h}{9} < \frac{79}{9} ] Simplifying, we get: [ h < \frac{79}{9} ]

Step 3: Simplify the Fraction

The fraction 79/9 is already in its simplest form, but it can be expressed as a mixed number for better understanding: [ \frac{79}{9} = 8 \frac{7}{9} ] So, the solution can also be written as: [ h < 8 \frac{7}{9} ]

Step 4: Interpret the Solution

The solution h < 79/9 means that h can take any value less than 79/9, but not equal to or greater than it. This includes all real numbers smaller than approximately 8.78.

Step 5: Graphical Representation (Optional)

On a number line, this solution would be represented by an open circle at 79/9 (or 8 7/9) and a line extending to the left, indicating all values less than this point.

Common Mistakes to Avoid

• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number (not applicable here since we divided by a positive 9).
• Misinterpreting the solution as a single value instead of a range.

Real-World Application

Inequalities like this are often used in real-life scenarios, such as budgeting (spending less than a certain amount) or time management (completing a task in less than a given time).

Conclusion

Solving the inequality 9h < 79 involves dividing both sides by 9 to isolate h, resulting in h < 79/9. This solution represents all real numbers less than 79/9, providing a clear range of possible values for h. Understanding how to solve such inequalities is fundamental in algebra and has practical applications in various fields.