Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all values of \(x\) such that \(f(x)=0.\) $$f(x)=\frac{9 x-4}{5}$$

Step-by-Step Solution

Answer

The value of \(x\) that makes \(f(x) = 0\) is \(x = \frac{4}{9}.\)

Step 1: Set the Function to Zero

Set \(f(x)\) equal to zero and write down the equation: \(0 = \frac{9x - 4}{5}.\)

Step 2: Simplify Equation

To get \(x\) alone, we first multiply both sides of the equation by 5, to get rid of the fraction: \(5 \cdot 0 = 5 \cdot \frac{9x - 4}{5} \rightarrow 0 = 9x - 4.\)

Step 3: Isolate Variable

Then, we add 4 to both sides of the equation to isolate \(x\): \(0 + 4 = 9x - 4 + 4 \rightarrow 4 = 9x.\)

Step 4: Solve for x

Finally, divide both sides of the equation by 9 to find the root: \(\frac{4}{9} = x. \)

Key Concepts

Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator, the denominator, or both. Understanding rational expressions is crucial in algebra, as they appear frequently in various contexts, such as solving equations, simplifying expressions, and analyzing functions.

To solve an equation involving a rational expression, like in the exercise with the function f(x) = \(\frac{9x - 4}{5}\), we look for the value of the variable that makes the entire expression equal to zero. This often involves clearing the fraction by multiplying both sides of the equation by the denominator. This step simplifies the rational expression into a polynomial, making it easier to find the root or solution.

Isolating Variables

Isolating the variable is a fundamental technique in algebra, particularly when solving linear equations. The goal is to get the variable by itself on one side of the equation. In other words, you want to extract the variable from among the other numbers and expressions.

For instance, the solution step where we add 4 to both sides of the equation, \(0 + 4 = 9x - 4 + 4\), is done to isolate the variable x on one side. Subsequent simplification yields \(4 = 9x\). Through these operations, we can manipulate an equation step by step to eventually isolate x and solve for its value.

Root Finding

Root finding, in the context of algebra, involves identifying the value or values of a variable for which a given equation or function equals zero. In the case of our exercise, finding the root means determining which value of x makes f(x) be zero.

To find this root, we first transform the rational expression into a simpler form where the variable is easily identifiable and can be isolated, as previously described. Once we have an equation with the variable on one side, like \(4 = 9x\), we can perform standard arithmetic operations to find the value of x that satisfies the equation, giving us the root of the function.

Algebraic Fractions

Algebraic fractions, also known as fractional expressions, are fractions that contain variables in their numerators, denominators, or both. They operate under the same principles as numeric fractions, but they can seem more challenging because of the presence of variables.

When encountering an algebraic fraction in an equation, such as \(\frac{9x - 4}{5}\), you treat the variables just like numbers. Simplifying algebraic fractions often involves finding a common denominator, reducing terms, and sometimes factoring polynomials. In solving equations, the common method is to eliminate the fraction entirely by multiplying both sides of the equation by the denominator. This aids in transforming the equation into a standard linear form, which is simpler to solve.