Problem 7 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each value of \(x\) is a solution of the equation. Values (a) \(x=-\frac{1}{2}\) (b) \(x=4\) (c) \(x=0\) (d) \(x=\frac{1}{4}\) Equation $$\frac{5}{2 x}-\frac{4}{x}=3$$

Step-by-Step Solution

Answer

Out of the given choices, only \(x=-\frac{1}{2}\) is a solution of the equation.

Step 1: Substitute the value \(x=-\frac{1}{2}\)

Replace \(x\) in the equation with \(-\frac{1}{2}\) and check the equality.\n The resulting equation is \(\frac{5}{2*(-\frac{1}{2})}-\frac{4}{-*\frac{1}{2}} = -5+8 = 3 \). So, \(x=-\frac{1}{2}\) is a solution.

Step 2: Substitute the value \(x=4\)

Replace \(x\) in the equation with \(4\) and check the equality.\n The resulting equation is \(\frac{5}{2*4}-\frac{4}{4} = \frac{5}{8}-1 = -\frac{3}{8} \), which is not equal to 3 . So, \(x=4\) is not a solution.

Step 3: Substitute the value \(x=0\)

If we replace \(x\) in the equation with \(0\), the expression becomes undefined due to zero in the denominator. So, \(x=0\) is not a solution.

Step 4: Substitute the value \(x=\frac{1}{4}\)

Replace \(x\) in the equation with \(\frac{1}{4}\) and check the equality.\n The resulting equation is \(\frac{5}{2*(\frac{1}{4})}-\frac{4}{\frac{1}{4}} = 10 - 16 = -6\), which is not equal to 3. So, \(x=\frac{1}{4}\) is not a solution.

Key Concepts

Solutions of Equations

In mathematics, solving an equation means finding all possible values of the variable that make the equation true. The given exercise involves checking if specific values of \(x\) satisfy the equation \(\frac{5}{2x} - \frac{4}{x} = 3\). This type of equation is considered a rational equation, which involves fractions where the numerators or denominators contain the variable \(x\). To determine whether a particular value is a solution, substitute it into the equation for \(x\) and simplify the expression. If both sides of the equation are equal after simplification, the value is a solution. In our given equation, only the substitution of \(x = -\frac{1}{2}\) makes it true, as both sides equal 3. The other provided values do not satisfy the equation.

Substitution Method

The substitution method is a fundamental technique to solve equations. This method involves replacing the variable with a given value and simplifying the expression to check if equality holds.Here's how you use it:

• Take the equation \(\frac{5}{2x} - \frac{4}{x} = 3\).
• Choose a value to substitute for \(x\) (e.g., \(x = 4\)) and replace \(x\) with this value.
• Simplify the resulting fraction expression.
• If the left side equals the right side of the equation, the value is a solution; otherwise, it is not.

In our exercise, after substituting, only \(x = -\frac{1}{2}\) keeps the equation balanced.

Undefined Expressions

A critical situation in solving rational equations occurs when the expression becomes undefined. This happens primarily due to division by zero. If substituting a value into the equation results in a denominator of zero, the expression has no defined value, making the equation unsolvable for that particular value.In the given exercise, substituting \(x = 0\) causes division by zero in both terms \(\frac{5}{2x}\) and \(\frac{4}{x}\) because \(x\) is in the denominator. An undefined expression means \(x = 0\) cannot be considered a solution to the equation. Always check the denominator when working with rational equations to avoid undefined results.

Fraction Operations

Handling fractions is a key part of working with rational equations. Fractions represent parts of a whole and involve operations that require careful attention to numerators and denominators. To solve equations like \(\frac{5}{2x} - \frac{4}{x} = 3\), you need to:

• Understand common denominators and simplify the expression correctly.
• When subtracting fractions, make sure the denominators are the same before performing the subtraction.
• Always reduce fractions to their simplest form for clarity.

In the exercise, calculating \(\frac{5}{2*(-\frac{1}{2})}-\frac{4}{-\frac{1}{2}}\) requires these skills to confirm that this operation equals 3. Mastering fraction operations will significantly aid in solving rational equations efficiently.