Solving Inequalities Finding The Complete Set Of Values For X

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In this comprehensive article, we will delve into the process of finding the complete set of values for x that satisfy the given inequality:

$$\frac{1}{2}(2 x+3)-\frac{2}{3}(x+1) < 2 x$$

We will break down the solution step-by-step, ensuring a clear understanding of the algebraic manipulations involved and arrive at the correct answer. This exploration will enhance your problem-solving skills in mathematics, particularly in dealing with linear inequalities.

Understanding the Inequality

At its core, this problem involves solving a linear inequality. Linear inequalities are mathematical statements that compare two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving an inequality means finding the range of values for the variable that makes the inequality true. This differs from solving equations, where we typically look for specific values that satisfy the equality.

The given inequality involves fractions and multiple terms, requiring careful simplification and manipulation. Our goal is to isolate x on one side of the inequality to determine the range of values that satisfy it. This process involves applying algebraic principles such as distribution, combining like terms, and performing operations on both sides of the inequality while maintaining its validity.

Before we dive into the step-by-step solution, it's crucial to understand the underlying concepts of inequalities. Inequalities behave similarly to equations in many ways, but there's a critical difference: multiplying or dividing both sides by a negative number reverses the inequality sign. This is a fundamental rule that we must keep in mind throughout the solution process.

Step 1: Distribute and Simplify

The first step in solving the inequality is to distribute the constants outside the parentheses to the terms inside. This will help us eliminate the parentheses and make the inequality easier to work with. Distributing the constants allows us to combine like terms and simplify the expression.

$$\frac{1}{2}(2 x+3)-\frac{2}{3}(x+1) < 2 x$$

Distribute the ${\frac{1}{2}}$ and ${-\frac{2}{3}}$:

$$\frac{1}{2} * 2x + \frac{1}{2} * 3 - \frac{2}{3} * x - \frac{2}{3} * 1 < 2x$$

Simplify each term:

$$x + \frac{3}{2} - \frac{2}{3}x - \frac{2}{3} < 2x$$

Now, we combine the x terms and the constant terms on the left side of the inequality. This involves identifying the terms with x and the constant terms and then adding or subtracting them accordingly. Combining like terms is a crucial step in simplifying algebraic expressions and making them easier to manipulate.

Group the x terms and constant terms:

$$(x - \frac{2}{3}x) + (\frac{3}{2} - \frac{2}{3}) < 2x$$

Find a common denominator for the x terms (which is 3) and the constant terms (which is 6). This is necessary to add or subtract fractions. Finding a common denominator ensures that we are working with fractions that have the same size pieces, allowing for accurate calculations.

$$(\frac{3}{3}x - \frac{2}{3}x) + (\frac{9}{6} - \frac{4}{6}) < 2x$$

Combine the terms:

$$\frac{1}{3}x + \frac{5}{6} < 2x$$

Step 2: Isolate the Variable Term

Now that we have simplified the left side of the inequality, our next goal is to isolate the variable term. This means getting all the terms with x on one side of the inequality and the constant terms on the other side. To do this, we will subtract ${\frac{1}{3}x}$ from both sides of the inequality.

Subtract ${\frac{1}{3}x}$ from both sides:

$$\frac{1}{3}x + \frac{5}{6} - \frac{1}{3}x < 2x - \frac{1}{3}x$$

Simplify:

$$\frac{5}{6} < 2x - \frac{1}{3}x$$

Now, we combine the x terms on the right side. This involves finding a common denominator (which is 3) and subtracting the fractions.

Find a common denominator:

$$\frac{5}{6} < \frac{6}{3}x - \frac{1}{3}x$$

Combine the terms:

$$\frac{5}{6} < \frac{5}{3}x$$

Step 3: Solve for x

To finally solve for x, we need to isolate it completely. Since x is being multiplied by ${\frac{5}{3}}$, we will divide both sides of the inequality by ${\frac{5}{3}}$. Remember that dividing by a fraction is the same as multiplying by its reciprocal. This step will give us the range of values for x that satisfy the inequality.

Divide both sides by ${\frac{5}{3}}$ (or multiply by ${\frac{3}{5}}$):

$$\frac{5}{6} * \frac{3}{5} < \frac{5}{3}x * \frac{3}{5}$$

Simplify:

$$\frac{15}{30} < x$$

Reduce the fraction:

$$\frac{1}{2} < x$$

This can also be written as:

$$x > \frac{1}{2}$$

Therefore, the complete set of values of x that satisfy the inequality is x greater than ${\frac{1}{2}}$.

Conclusion

In this article, we have successfully navigated the process of solving a linear inequality. By carefully applying algebraic principles such as distribution, combining like terms, and performing operations on both sides of the inequality, we have arrived at the solution: x > ${\frac{1}{2}}$. This means that any value of x greater than ${\frac{1}{2}}$ will satisfy the original inequality.

Understanding how to solve inequalities is a fundamental skill in mathematics. It allows us to analyze and solve a wide range of problems, from simple algebraic expressions to more complex real-world scenarios. The key to success lies in breaking down the problem into manageable steps, carefully applying the rules of algebra, and paying close attention to the direction of the inequality sign.

By mastering the techniques discussed in this article, you will be well-equipped to tackle a variety of inequality problems and enhance your mathematical problem-solving abilities. Remember to practice regularly and apply these concepts to different scenarios to solidify your understanding. With dedication and perseverance, you can excel in mathematics and unlock its many exciting possibilities.

Therefore, the answer is A. x > ${\frac{1}{2}}$