Solve The Inequality 9h 2 79

Solve the Inequality 9h + 2 ≤ 79: A Step-by-Step Guide

Understanding how to solve linear inequalities is a fundamental skill in algebra that helps you compare quantities and find ranges of possible solutions. Worth adding: the inequality 9h + 2 ≤ 79 is a straightforward example that demonstrates the core principles of isolating variables and maintaining the balance of an inequality. This guide will walk you through the process of solving this inequality, explain the underlying mathematical reasoning, and provide practical insights to reinforce your understanding.

Steps to Solve the Inequality 9h + 2 ≤ 79

Solving the inequality 9h + 2 ≤ 79 involves a series of systematic steps designed to isolate the variable h. Follow these steps carefully:

1. Subtract 2 from both sides of the inequality:
   Begin by eliminating the constant term on the left side. Subtract 2 from both sides to maintain the balance:

   9h + 2 - 2 ≤ 79 - 2  
   9h ≤ 77
   

2. Divide both sides by 9:
   Next, isolate h by dividing both sides of the inequality by the coefficient 9:

   9h ÷ 9 ≤ 77 ÷ 9  
   h ≤ 8.555...
   

   Simplifying the right side, we get h ≤ 8.555... or h ≤ 77/9 in fractional form.

3. Express the solution clearly:
   The final result shows that h can be any number less than or equal to 77/9 (approximately 8.56). In interval notation, this is written as (-∞, 77/9].

By following these steps, you’ve successfully isolated the variable and determined the range of values that satisfy the inequality.

Scientific Explanation: Why These Steps Work

The process of solving inequalities relies on the properties of inequality, which are similar to the properties of equality but with a few critical differences. Here’s why each step is valid:

• Subtraction Property of Inequality: When you subtract the same number from both sides of an inequality, the direction of the inequality remains unchanged. This property ensures that the relationship between the two sides is preserved.

• Division Property of Inequality: When dividing both sides of an inequality by a positive number, the inequality sign stays the same. Even so, if you were dividing by a negative number, the inequality sign would reverse. In this case, since we divided by 9 (a positive number), the direction of the inequality remains ≤.

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These properties are rooted in the fundamental behavior of numbers and operations. Here's the thing — for example, if a ≤ b, then adding or subtracting the same value from both sides maintains the relationship. Think about it: similarly, multiplying or dividing by a positive number preserves the inequality, while multiplying or dividing by a negative number reverses it. Understanding these principles ensures that your solutions are mathematically sound.

Frequently Asked Questions (FAQ)

1. What happens if I multiply or divide by a negative number?

If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. Here's one way to look at it: if you had -9h ≤ -77, dividing by -9 would yield h ≥ 8.555.... This rule is crucial to avoid incorrect solutions.

2. How do I represent the solution graphically?

On a number line, the solution h ≤ 77/9 is represented by a closed circle at 77/9 (since the inequality includes equality) and a shaded line extending to the left (toward negative infinity). This visual representation helps you see the range of valid values at a glance.

3. Can I use equations to solve inequalities?

While the steps for solving inequalities are similar to those for equations, the key difference lies in the inequality sign. You can use the same algebraic operations (addition, subtraction, multiplication, division) as long as you respect the rules for inequality signs. Always double-check whether the inequality sign needs to be reversed.

4. What is the practical use of inequalities?

Inequalities are widely used in real-world scenarios, such as budgeting (e.g., ensuring expenses do not exceed income), engineering (e.g., stress limits on materials), and economics (e.g., profit margins). Here's one way to look at it: if you’re planning a event and need to stay within a budget, an inequality like cost ≤ budget can help you determine feasible options.

Conclusion

Solving the inequality 9h + 2 ≤ 79 is a foundational exercise in algebra that reinforces the principles of equation balancing and inequality properties. Day to day, by systematically subtracting 2 and dividing by 9, you isolate the variable and determine the solution h ≤ 77/9. Mastering these steps not only helps you solve similar problems but also builds a strong foundation for more advanced topics in mathematics.

and changing the constants, to solidify your understanding. Remember to always pay close attention to the direction of the inequality sign when performing operations – this is the key to accurate and reliable solutions. To build on this, visualizing the solution on a number line provides a valuable tool for confirming your answer and grasping the full scope of the possible values. Because of that, finally, recognizing the practical applications of inequalities demonstrates their importance beyond the classroom, highlighting their relevance in diverse fields. Consistent application of these rules and techniques will undoubtedly improve your ability to tackle a wide range of algebraic problems and confidently interpret mathematical relationships.