Is 0 A Solution Of -1/2 ≥ -3/4 + X? Explained!

Let's dive into the question: Is the number 0 part of the solution set for the inequality $-\frac{1}{2} \geq -\frac{3}{4} + x$?

Understanding the Inequality

First, we need to understand what this inequality means. The inequality $-\frac{1}{2} \geq -\frac{3}{4} + x$ is a mathematical statement that says "negative one-half is greater than or equal to negative three-fourths plus x." Our mission is to determine whether substituting 0 for x makes this statement true.

To figure this out, we'll substitute 0 for x in the inequality and see if the resulting statement holds. This involves basic arithmetic and a clear understanding of inequality rules. Let's break it down step by step to ensure we understand each part of the process.

Step-by-Step Solution

1. Substitute 0 for x: Replace x with 0 in the inequality.

   • $-\frac{1}{2} \geq -\frac{3}{4} + 0$

2. Simplify the Inequality: Since adding 0 to any number doesn't change the number, we can simplify the right side.

   • $-\frac{1}{2} \geq -\frac{3}{4}$

3. Compare the Fractions: Now we need to determine if $-\frac{1}{2}$ is indeed greater than or equal to $-\frac{3}{4}$. To make this comparison easier, we can find a common denominator for both fractions. The least common denominator for 2 and 4 is 4.

   • Convert $-\frac{1}{2}$ to a fraction with a denominator of 4: $-\frac{1}{2} = -\frac{2}{4}$

4. Rewrite the Inequality: Substitute $-\frac{1}{2}$ with $-\frac{2}{4}$ in the inequality.

   • $-\frac{2}{4} \geq -\frac{3}{4}$

5. Determine if the Inequality Holds: Now we compare the two fractions. Remember, when dealing with negative numbers, the number closer to zero is greater. Is $-\frac{2}{4}$ greater than or equal to $-\frac{3}{4}$?

Comparing Negative Fractions

Think of a number line. $-\frac{2}{4}$ is to the right of $-\frac{3}{4}$, meaning it is closer to zero and thus greater. Therefore, $-\frac{2}{4} \geq -\frac{3}{4}$ is a true statement.

Conclusion

Since the inequality $-\frac{1}{2} \geq -\frac{3}{4} + x$ holds true when we substitute 0 for x, the number 0 is part of the solution set for the inequality. Therefore, option A is the correct answer.

Why Other Options Are Incorrect

It's also helpful to understand why the other options are incorrect:

• Option B: No, because the number 0 is not part of the horizontal number line.
  • This statement is factually incorrect. The number 0 is definitely part of the horizontal number line. It sits right in the middle, separating the positive and negative numbers. So, this can't be the reason why 0 might not be a solution.
• Option C: [Incomplete]
  • Without the full statement of option C, it’s hard to address it specifically. However, the key to these types of problems is always to substitute and check if the inequality holds true.

Key Concepts in Solving Inequalities

When solving inequalities, several key concepts can help you arrive at the correct solution efficiently and accurately. Understanding these concepts ensures that you're not just memorizing steps but truly grasping the underlying principles.

Substitution

Substitution is a fundamental technique in algebra. It involves replacing a variable with a specific value to evaluate an expression or check if an equation or inequality holds true. In the context of inequalities, substitution helps determine whether a particular number is part of the solution set.

• How to Use Substitution: To use substitution, simply replace the variable (in this case, x) with the given number (0). Then, simplify the inequality to see if the resulting statement is true. If the statement is true, the number is part of the solution set; otherwise, it is not.
• Example: Consider the inequality $x + 2 < 5$. To check if 1 is a solution, substitute x with 1: $1 + 2 < 5$, which simplifies to $3 < 5$. Since this is true, 1 is part of the solution set.

Common Denominators

When dealing with inequalities (or equations) involving fractions, finding a common denominator is often necessary to compare or combine the fractions. A common denominator is a number that is a multiple of the denominators of all the fractions involved.

• Why Use Common Denominators? Using a common denominator allows you to compare fractions directly. It ensures that you are comparing like quantities, making it easier to determine which fraction is larger or smaller.
• How to Find a Common Denominator: The least common denominator (LCD) is usually the easiest to work with. To find the LCD, identify the smallest number that each denominator can divide into evenly. For example, if you have fractions with denominators 2, 3, and 4, the LCD is 12 because 12 is the smallest multiple of 2, 3, and 4.
• Example: To compare $-\frac{1}{2}$ and $-\frac{3}{4}$, find the common denominator, which is 4. Convert $-\frac{1}{2}$ to $-\frac{2}{4}$. Now you can easily compare $-\frac{2}{4}$ and $-\frac{3}{4}$.

Number Line Visualization

Visualizing numbers on a number line is a powerful tool for understanding inequalities, especially when dealing with negative numbers. A number line provides a visual representation of the order and relative position of numbers.

• How to Use a Number Line: Draw a horizontal line and mark zero in the middle. Positive numbers are to the right of zero, and negative numbers are to the left. The farther a number is to the right, the greater it is. Conversely, the farther a number is to the left, the smaller it is.
• Example: To compare $-\frac{2}{4}$ and $-\frac{3}{4}$, visualize their positions on the number line. $-\frac{2}{4}$ is to the right of $-\frac{3}{4}$, indicating that $-\frac{2}{4}$ is greater than $-\frac{3}{4}$.

Rules for Negative Numbers

When working with inequalities, understanding the rules for negative numbers is crucial. Negative numbers behave differently than positive numbers, especially when it comes to comparing their values.

• Comparing Negative Numbers: The closer a negative number is to zero, the greater its value. For example, -1 is greater than -2 because -1 is closer to zero on the number line.
• Multiplying or Dividing by a Negative Number: When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. For example, if $x < -2$, then multiplying both sides by -1 gives $-x > 2$.
• Example: Comparing $-\frac{1}{2}$ and $-\frac{3}{4}$: $-\frac{1}{2}$ is equivalent to $-\frac{2}{4}$. Since $-\frac{2}{4}$ is closer to zero than $-\frac{3}{4}$, $-\frac{2}{4}$ is greater than $-\frac{3}{4}$.

Simplifying Inequalities

Simplifying inequalities involves performing operations to isolate the variable and make the inequality easier to understand. The goal is to get the variable by itself on one side of the inequality sign.

• Basic Operations: You can add, subtract, multiply, or divide both sides of an inequality by the same number, with one important exception: multiplying or dividing by a negative number (as mentioned above).
• Combining Like Terms: Combine like terms on each side of the inequality to simplify it. For example, in the inequality $2x + 3 - x < 7$, combine $2x$ and $-x$ to get $x + 3 < 7$.
• Example: To solve the inequality $-\frac{1}{2} \geq -\frac{3}{4} + x$, you can add $\frac{3}{4}$ to both sides: $-\frac{1}{2} + \frac{3}{4} \geq x$. Simplify the left side to get $\frac{1}{4} \geq x$, which means $x \leq \frac{1}{4}$.

Real-World Applications

Understanding inequalities isn't just about solving math problems; it has numerous real-world applications. Inequalities are used in various fields to model and solve problems involving constraints, comparisons, and optimization.

Budgeting and Finance

Inequalities are commonly used in budgeting and finance to set limits and manage expenses. For example, if you have a budget of $500 per month, you can use the inequality $expenses \leq 500$ to ensure that your total expenses do not exceed your budget.

• Example: Suppose you want to save at least $100 each month. You can express this as $savings \geq 100$. If your income is $2000 per month, and you want to ensure your expenses and savings stay within your income, you can use the inequality $expenses + savings \leq 2000$.

Engineering

In engineering, inequalities are used to set tolerance limits and ensure that designs meet certain specifications. For example, the diameter of a pipe must be within a certain range to ensure proper flow.

• Example: A bridge must be able to withstand a certain load. The inequality $load \leq maxLoad$ ensures that the actual load on the bridge does not exceed the maximum load it can safely handle. Similarly, the temperature of a chemical reaction must be kept within a specific range, expressed as $minTemp \leq temperature \leq maxTemp$, to prevent dangerous conditions.

Computer Science

Inequalities are used in computer science for algorithm design, optimization, and resource allocation. For example, the running time of an algorithm must be less than a certain threshold to ensure it is efficient.

• Example: The memory usage of a program must be less than the available memory, expressed as $memoryUsed \leq memoryAvailable$. In network routing, the delay must be minimized, and the inequality $delay \leq maxDelay$ ensures that the delay does not exceed a certain limit.

Health and Medicine

In health and medicine, inequalities are used to set health parameters and monitor patient conditions. For example, a patient's blood pressure must be within a certain range to ensure they are healthy.

• Example: A person's BMI (Body Mass Index) should be within a healthy range, expressed as $18.5 \leq BMI \leq 24.9$. Similarly, blood sugar levels must be maintained within a certain range, expressed as $minBloodSugar \leq bloodSugar \leq maxBloodSugar$, to prevent diabetes-related complications.

Everyday Life

Inequalities are also used in everyday life for decision-making and problem-solving. Whether you're planning a trip, managing your time, or comparing prices, inequalities can help you make informed choices.

• Example: If you have to travel a certain distance in a limited time, you can use the inequality $speed \geq minSpeed$ to ensure you arrive on time. When shopping, you can compare prices using the inequality $priceA \leq priceB$ to find the best deal.

In summary, inequalities are a fundamental tool in mathematics with wide-ranging applications across various fields. By understanding the key concepts and techniques for solving inequalities, you can tackle a wide range of problems and make informed decisions in real-world situations.

Conclusion

In conclusion, by substituting 0 for x in the inequality $-\frac{1}{2} \geq -\frac{3}{4} + x$ and simplifying, we found that the inequality holds true. Therefore, the number 0 is indeed part of the solution set. Remembering these steps and practicing similar problems will help you master inequalities. For further learning, visit Khan Academy's page on inequalities.