Solving Inequalities The Correct First Step

In the realm of mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Unlike equations that seek exact solutions, inequalities express a range of possible values that satisfy a given condition. Solving inequalities involves isolating the variable of interest, similar to solving equations, but with a few key differences to keep in mind. This comprehensive guide will delve into the intricacies of solving inequalities, focusing specifically on identifying the correct first step in solving the inequality $-4(3-5x) ext{≥} -6x + 9$.

Understanding Inequalities

Inequalities are mathematical statements that compare two expressions using inequality symbols such as "greater than" (>), "less than" (<), "greater than or equal to" (≥), and "less than or equal to" (≤). These symbols indicate that the values on either side of the inequality are not necessarily equal. For example, the inequality $x > 5$ signifies that the value of $x$ is greater than 5, but not equal to 5. Similarly, $y ≤ 10$ means that $y$ is less than or equal to 10.

Solving inequalities involves finding the set of values that satisfy the inequality. This set of values is often represented as an interval on the number line. For instance, the solution to $x > 5$ is the interval $(5, ∞)$, which includes all values greater than 5. The solution to $y ≤ 10$ is the interval $(-∞, 10]$, which includes all values less than or equal to 10.

The Distributive Property: A Crucial First Step

When tackling inequalities, especially those involving parentheses, the distributive property often serves as the initial key to unlocking the solution. The distributive property states that multiplying a number by a sum or difference is the same as multiplying the number by each term inside the parentheses individually and then adding or subtracting the results. Mathematically, this can be expressed as:

$a(b + c) = ab + ac$

$a(b - c) = ab - ac$

Applying the distributive property correctly is crucial for simplifying inequalities and paving the way for isolating the variable. Let's consider the inequality $-4(3-5x) ext{≥} -6x + 9$. The first step in solving this inequality involves distributing the -4 across the terms inside the parentheses.

Applying the Distributive Property to the Inequality

In the given inequality, $-4(3-5x) ext{≥} -6x + 9$, we need to distribute the -4 to both the 3 and the -5x terms within the parentheses. Let's break down this process step by step:

1. Multiply -4 by 3: $-4 * 3 = -12$
2. Multiply -4 by -5x: $-4 * -5x = 20x$

Now, we can rewrite the left side of the inequality by combining these results:

$-4(3 - 5x) = -12 + 20x$

Therefore, the inequality becomes:

$-12 + 20x ext{≥} -6x + 9$

This transformation represents the correct first step in solving the inequality. By applying the distributive property, we have eliminated the parentheses and simplified the expression, making it easier to proceed with isolating the variable x.

Analyzing the Options

Now, let's examine the given options and identify the one that matches our result:

A. $-12 - 20x ext{≤} -6x + 9$

This option incorrectly distributes the -4, resulting in a negative sign for the 20x term. The correct distribution should yield a positive 20x.

B. $-12 - 20x ext{≥} -6x + 9$

Similar to option A, this option also incorrectly distributes the -4, leading to a negative 20x term instead of a positive one.

C. $-12 + 20x ext{≤} -6x + 9$

This option correctly distributes the -4, resulting in -12 + 20x. However, it incorrectly flips the inequality sign. When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed. In this case, we are only distributing a negative number, not multiplying both sides of the inequality by a negative number; therefore, the inequality sign should remain the same.

D. $-12 + 20x ext{≥} -6x + 9$

This option accurately distributes the -4, resulting in -12 + 20x, and it correctly maintains the original inequality sign (≥). This is the correct first step in solving the inequality.

The Correct First Step: Option D

Based on our analysis, the correct first step in solving the inequality $-4(3-5x) ext{≥} -6x + 9$ is:

D. $-12 + 20x ext{≥} -6x + 9$

This step involves correctly applying the distributive property to eliminate the parentheses and simplify the inequality.

Continuing the Solution

Once we have taken the correct first step, we can proceed with solving the inequality by isolating the variable x. The subsequent steps involve:

1. Combining like terms: Add 6x to both sides of the inequality to get the x terms on one side:

   $-12 + 20x + 6x ext{≥} -6x + 9 + 6x$

   $-12 + 26x ext{≥} 9$

2. Isolating the variable term: Add 12 to both sides to isolate the term with x:

   $-12 + 26x + 12 ext{≥} 9 + 12$

   $26x ext{≥} 21$

3. Solving for x: Divide both sides by 26 to solve for x:

   rac{26x}{26} ext{≥} rac{21}{26}

   x ext{≥} rac{21}{26}

Therefore, the solution to the inequality is x ext{≥} rac{21}{26}. This means that any value of x greater than or equal to rac{21}{26} will satisfy the original inequality.

Key Takeaways

• The distributive property is a fundamental tool for simplifying expressions and inequalities involving parentheses.
• Careful application of the distributive property is crucial for avoiding errors in sign and coefficient calculations.
• Understanding the rules for manipulating inequalities is essential for arriving at the correct solution. Remember to reverse the inequality sign when multiplying or dividing both sides by a negative number.
• Solving inequalities involves isolating the variable to determine the range of values that satisfy the given condition.

By mastering these concepts and techniques, you can confidently tackle a wide range of inequality problems.

Conclusion

In conclusion, the correct first step in solving the inequality $-4(3-5x) ext{≥} -6x + 9$ is to apply the distributive property, which leads to the equivalent inequality $-12 + 20x ext{≥} -6x + 9$. This step simplifies the inequality and sets the stage for further manipulation to isolate the variable and find the solution. Remember, a solid understanding of the distributive property and the rules for manipulating inequalities is key to success in solving these types of problems. By following a systematic approach and paying close attention to detail, you can confidently navigate the world of inequalities and achieve accurate solutions.