Solving Inequalities Step-by-Step Guide To 9h + 2 Less Than -79

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In the realm of mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Unlike equations that assert equality, inequalities express the relative order of two expressions. Mastering the techniques for solving inequalities is fundamental for various mathematical applications, including optimization problems, calculus, and real-world scenarios involving constraints and limitations. This comprehensive guide delves into the step-by-step process of solving the inequality $9h + 2 < -79$, providing a clear understanding of the underlying principles and practical applications.

Understanding Inequalities

Before diving into the specifics of solving $9h + 2 < -79$, it's essential to grasp the fundamental concepts of inequalities. Inequalities use symbols to compare values, indicating whether one value is less than, greater than, less than or equal to, or greater than or equal to another value. The primary symbols used in inequalities are:

  • : < (less than): Indicates that one value is smaller than another.
  • : > (greater than): Indicates that one value is larger than another.
  • : ≤ (less than or equal to): Indicates that one value is smaller than or equal to another.
  • : ≥ (greater than or equal to): Indicates that one value is larger than or equal to another.

Inequalities can represent a range of values, unlike equations that typically have a finite set of solutions. This range of values is often depicted graphically on a number line, providing a visual representation of the solution set.

Solving the Inequality $9h + 2 < -79$

Now, let's tackle the inequality $9h + 2 < -79$ step by step. The goal is to isolate the variable h on one side of the inequality to determine the range of values that satisfy the condition.

Step 1: Isolate the Term with the Variable

The first step involves isolating the term containing the variable h. In this case, we need to get rid of the constant term +2 on the left side of the inequality. To do this, we apply the inverse operation, which is subtraction. We subtract 2 from both sides of the inequality:

9h+2−2<−79−29h + 2 - 2 < -79 - 2

This simplifies to:

9h<−819h < -81

Step 2: Isolate the Variable

Now that we have isolated the term with the variable, we need to isolate h itself. The variable h is currently being multiplied by 9. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the inequality by 9:

9h9<−819\frac{9h}{9} < \frac{-81}{9}

This simplifies to:

h<−9h < -9

Solution

The solution to the inequality $9h + 2 < -79$ is $h < -9$. This means that any value of h that is less than -9 will satisfy the original inequality. For example, if we substitute h = -10 into the original inequality, we get:

9(−10)+2<−799(-10) + 2 < -79

−90+2<−79-90 + 2 < -79

−88<−79-88 < -79

This statement is true, confirming that h = -10 is a valid solution. On the other hand, if we substitute h = -8 into the original inequality, we get:

9(−8)+2<−799(-8) + 2 < -79

−72+2<−79-72 + 2 < -79

−70<−79-70 < -79

This statement is false, indicating that h = -8 is not a solution.

Graphical Representation

The solution $h < -9$ can be represented graphically on a number line. We draw an open circle at -9 to indicate that -9 is not included in the solution set (since the inequality is strictly less than). Then, we shade the region to the left of -9 to represent all values of h that are less than -9. This shaded region visually represents the solution set of the inequality.

Key Concepts and Rules for Solving Inequalities

Solving inequalities involves applying similar principles to solving equations, but there are a few key differences to keep in mind:

  • Addition and Subtraction: Adding or subtracting the same value from both sides of an inequality does not change the direction of the inequality sign.
  • Multiplication and Division by a Positive Number: Multiplying or dividing both sides of an inequality by a positive number does not change the direction of the inequality sign.
  • Multiplication and Division by a Negative Number: Multiplying or dividing both sides of an inequality by a negative number does change the direction of the inequality sign. This is a crucial rule to remember when working with inequalities.
  • Flipping the Inequality: If you multiply or divide by a negative number, you must flip the inequality sign. For example, if you have -2x < 6, dividing by -2 would result in x > -3.

Common Mistakes to Avoid

When solving inequalities, it's important to avoid common mistakes that can lead to incorrect solutions. Some of the most frequent errors include:

  • Forgetting to Flip the Inequality Sign: This is the most common mistake when multiplying or dividing both sides of an inequality by a negative number. Always remember to reverse the direction of the inequality sign in such cases.
  • Incorrectly Applying the Distributive Property: When dealing with inequalities involving parentheses, ensure that you correctly apply the distributive property to avoid errors.
  • Combining Like Terms Incorrectly: Combine like terms carefully to simplify the inequality before proceeding with further steps.
  • Misinterpreting the Solution Set: Understand the meaning of the inequality symbols and correctly interpret the solution set. For example, x < 5 means all values less than 5, but not including 5 itself.

Real-World Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have numerous practical applications in various real-world scenarios. Here are a few examples:

  • Budgeting: Inequalities can be used to represent budget constraints. For instance, if you have a budget of $100, you can use the inequality x + y ≤ 100 to represent the possible amounts you can spend on two different items, where x and y are the costs of the items.
  • Optimization Problems: Inequalities play a vital role in optimization problems, where the goal is to find the maximum or minimum value of a function subject to certain constraints. These constraints are often expressed as inequalities.
  • Tolerance and Error Analysis: In engineering and manufacturing, inequalities are used to specify tolerance limits and error bounds. For example, the dimensions of a manufactured part might need to fall within a certain range, which can be expressed as an inequality.
  • Health and Fitness: Inequalities can be used to represent healthy ranges for various health indicators, such as body mass index (BMI), blood pressure, and cholesterol levels.
  • Speed Limits: Speed limits on roads are a practical application of inequalities. For example, a speed limit of 65 mph can be expressed as v ≤ 65, where v is the vehicle's speed.

Advanced Inequality Techniques

While the basic principles of solving inequalities are relatively straightforward, some inequalities require more advanced techniques. These include:

Compound Inequalities

Compound inequalities involve two or more inequalities connected by the words "and" or "or." For example, the compound inequality "-3 < x ≤ 5" represents all values of x that are greater than -3 and less than or equal to 5. To solve compound inequalities, you need to solve each inequality separately and then combine the solutions according to the connecting word.

Absolute Value Inequalities

Absolute value inequalities involve the absolute value of an expression. The absolute value of a number is its distance from zero, regardless of direction. For example, |x| < 3 means that the distance between x and 0 is less than 3, which is equivalent to the compound inequality -3 < x < 3. Solving absolute value inequalities requires careful consideration of the different cases that arise due to the absolute value.

Quadratic Inequalities

Quadratic inequalities involve quadratic expressions (expressions with a term raised to the power of 2). To solve quadratic inequalities, you typically need to factor the quadratic expression, find the critical points (the values that make the expression equal to zero), and then test intervals on the number line to determine the solution set.

Conclusion

Solving inequalities is a fundamental skill in mathematics with wide-ranging applications in various fields. By understanding the basic principles, following the correct steps, and avoiding common mistakes, you can confidently solve a wide range of inequalities. Remember to pay close attention to the direction of the inequality sign, especially when multiplying or dividing by a negative number. With practice and a solid understanding of the concepts, you can master the art of solving inequalities and apply them to real-world problems.

In this comprehensive guide, we've explored the step-by-step process of solving the inequality $9h + 2 < -79$, along with a broader discussion of inequalities, key concepts, common mistakes, real-world applications, and advanced techniques. This knowledge will empower you to tackle more complex mathematical challenges and apply inequalities effectively in various contexts.

Choosing the Correct Answer

Based on our step-by-step solution, we found that $h < -9$. Therefore, the correct answer is:

C. $h < -9$