Solving Inequalities A Step-by-Step Guide To 9h + 2 < -79

Understanding Inequalities

Before diving into solving the specific inequality 9h + 2 < -79, let's first understand what inequalities are and how they differ from equations. Inequalities, in mathematics, are mathematical expressions that show the relationship between two values that are not necessarily equal. Unlike equations, which use an equals sign (=) to show that two expressions are equivalent, inequalities use symbols such as less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥) to indicate the relative order of values. Solving an inequality means finding the range of values that satisfy the inequality condition, rather than a single solution as in an equation. The basic principles for solving inequalities are similar to those for solving equations, but with one crucial difference: when multiplying or dividing both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the sign of the values, and thus their order on the number line. For example, if 2 < 4, multiplying both sides by -1 gives -2 and -4. However, -2 is greater than -4, so the inequality sign must be reversed to -2 > -4. Grasping this concept is vital for accurately solving inequalities. Inequalities are used extensively in various fields, from economics and physics to computer science, to model and analyze situations where precise equality is not required or is not possible. For instance, in economics, inequalities can represent budget constraints or profit margins. In physics, they might describe the range of possible values for a physical quantity. This makes the ability to solve inequalities a fundamental skill in many quantitative disciplines.

Step-by-Step Solution

To solve the inequality 9h + 2 < -79, we'll follow a step-by-step approach, similar to solving equations, but with careful attention to the direction of the inequality sign. Our primary goal is to isolate the variable h on one side of the inequality. The first step involves eliminating the constant term (+2) from the left side of the inequality. We can achieve this by subtracting 2 from both sides of the inequality. This maintains the balance of the inequality, just as subtracting the same value from both sides of an equation does. So, we perform the operation: 9h + 2 - 2 < -79 - 2. This simplifies to 9h < -81. Now, we need to isolate h by removing the coefficient 9. Since 9 is multiplying h, we perform the inverse operation, which is division. We divide both sides of the inequality by 9: (9h) / 9 < (-81) / 9. This simplifies to h < -9. An important point to note here is that we divided by a positive number (9), so we do not need to reverse the inequality sign. If we were dividing by a negative number, we would need to flip the inequality sign to maintain the correct relationship. The solution, h < -9, means that any value of h that is less than -9 will satisfy the original inequality. This is not a single value, but rather a range of values, extending infinitely in the negative direction. Representing solutions to inequalities often involves graphing them on a number line, which visually illustrates the range of acceptable values. In this case, the number line would show an open circle at -9 (since h is strictly less than -9 and not equal to -9) and a line extending to the left, indicating all values less than -9 are solutions.

Evaluating the Options

Now that we've solved the inequality 9h + 2 < -79 and found the solution to be h < -9, we can evaluate the given options to determine which one is correct. The options provided are:

A. h < 9 B. h < 10 C. h < -9

Comparing our solution (h < -9) to the options, it becomes clear that option C, h < -9, is the correct answer. Option A (h < 9) and option B (h < 10) are incorrect because they represent values of h that are greater than -9, which do not satisfy the original inequality. To further illustrate this, let's consider a few examples. If we were to substitute h = -10 (which is less than -9) into the original inequality, we would get: 9(-10) + 2 < -79, which simplifies to -90 + 2 < -79, and further to -88 < -79. This statement is true, so h = -10 is a valid solution. On the other hand, if we were to substitute h = 0 (which is greater than -9) into the original inequality, we would get: 9(0) + 2 < -79, which simplifies to 2 < -79. This statement is false, so h = 0 is not a solution. This process of substituting values into the original inequality is a useful way to check your solution and ensure it is correct. It reinforces the understanding that an inequality represents a range of values, and only values within that range will satisfy the condition. In conclusion, option C is the only option that accurately represents the solution set for the inequality 9h + 2 < -79.

Common Mistakes and How to Avoid Them

Solving inequalities, like solving equations, requires careful attention to detail to avoid common mistakes. One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This mistake can completely change the solution set. For example, if we had an inequality like -2x < 6, dividing both sides by -2 requires flipping the inequality sign, resulting in x > -3. Failing to do so would incorrectly yield x < -3. To avoid this, always double-check if you're multiplying or dividing by a negative number and make sure to reverse the inequality sign if necessary. Another common mistake is incorrectly applying the order of operations. When simplifying inequalities, it's crucial to follow the correct order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Ignoring this order can lead to incorrect simplifications and a wrong solution. For instance, in the inequality 3(x + 2) > 9, you should first distribute the 3 across the parentheses before attempting to isolate x. A third common error is misinterpreting the solution set. Remember that the solution to an inequality is often a range of values, not a single value. It's important to understand what the inequality sign means and how it translates to the solution set. For example, x > 5 means all values greater than 5, but not including 5 itself. Representing the solution set on a number line can help visualize the range of values that satisfy the inequality. Finally, always check your solution by substituting a value from your solution set back into the original inequality to ensure it holds true. This simple step can catch many errors and give you confidence in your answer. By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy in solving inequalities.

Alternative Approaches

While the step-by-step algebraic method is the standard approach to solving inequalities, there are alternative methods that can be useful in certain situations or for checking your work. One such method is the graphical approach. This involves plotting the expressions on both sides of the inequality as functions on a graph and then visually identifying the range of x-values where one function is greater than or less than the other. For example, in the inequality 9h + 2 < -79, we could graph the lines y = 9h + 2 and y = -79. The solution to the inequality would be the set of h-values where the line y = 9h + 2 is below the line y = -79. This graphical method provides a visual representation of the solution set and can be particularly helpful for understanding more complex inequalities or systems of inequalities. Another alternative approach is to use test values. This involves selecting a value within a potential solution range and substituting it back into the original inequality to see if it holds true. If the test value satisfies the inequality, then the range it belongs to is likely part of the solution set. For example, after solving 9h + 2 < -79 to get h < -9, we could test h = -10 (which is less than -9) in the original inequality. Since 9(-10) + 2 < -79 is true, this confirms that values less than -9 are indeed part of the solution. Conversely, testing h = 0 (which is not less than -9) would show that it does not satisfy the inequality. This method is particularly useful for checking your solution and identifying potential errors. A third approach, which is more of a mental math technique, is to try to estimate the solution. This involves thinking about what values of the variable would make the inequality true without going through all the algebraic steps. While this approach may not always be precise, it can help you get a sense of the solution and can be useful for multiple-choice questions where you can eliminate obviously incorrect options. For example, in 9h + 2 < -79, you might think, "9 times what number, plus 2, is less than -79?" This can lead you to realize that the number must be a negative number around -9. While these alternative approaches may not always be the most efficient for solving all inequalities, they provide valuable tools for understanding, visualizing, and checking your solutions.

Therefore, the correct answer is C. h < -9