Problem 53 - College Algebra

Question

Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact \(x\) -intercept between your right and left boundaries. With other types of functions (more than one \(x\) -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left nd right boundaries. \(\mathrm{Y}_{1}=\frac{3 x+5}{4}\) Round your answer to the nearest thousandth.

Step-by-Step Solution

Answer

The x-intercept is approximately \(x = -1.667\).

Step 1: Understand the Problem

The problem is asking us to find the x-intercept of the given function. The x-intercept is the point where the function crosses the x-axis, which means the function's output is zero at that point.

Step 2: Set the Function to Zero

Given the function \(Y_1 = \frac{3x + 5}{4}\), we need to find the value of \(x\) where \(Y_1 = 0\). Set the function equal to zero: \[0 = \frac{3x + 5}{4}\].

Step 3: Solve for x

Multiply both sides by 4 to eliminate the denominator: \[0 = 3x + 5\]. Next, isolate \(x\) by subtracting 5 from both sides: \[-5 = 3x\]. Finally, divide both sides by 3 to solve for \(x\): \[x = -\frac{5}{3}\].

Step 4: Convert the Answer to Decimal Form

Convert \(-\frac{5}{3}\) to a decimal. \(-\frac{5}{3} = -1.6666\ldots\). This is a repeating decimal, so round to the nearest thousandth: \(-1.667\).

Step 5: Validate the Solution

Check by substituting \(x = -1.667\) into the original equation to ensure the output is approximately zero: \(Y_1 = \frac{3(-1.667) + 5}{4} \approx 0\). The number should be very close to zero, confirming the solution.

Key Concepts

Understanding the X-Intercept

The x-intercept of a function is a fundamental concept in understanding how a graph interacts with the x-axis. It is specifically the point where the graph of the function crosses, or touches, the x-axis. This means that at these points, the value of the function, or the output, is zero. In mathematical terms, it is the value of \(x\) when \(f(x) = 0\).

• To find the x-intercept, you set the function equal to zero.
• This involves solving the equation \(f(x) = 0\) for \(x\).
• There can be one or multiple x-intercepts, especially in non-linear functions.

For the given linear function \(Y_1 = \frac{3x + 5}{4}\), finding the x-intercept involves setting the fraction equal to zero and solving for \(x\). This is because that is the value at which the graph will cross the x-axis, indicating the output of the function is zero.

Exploring Linear Functions

Linear functions are equations that create a straight line when graphed. They have the general form \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) represents the y-intercept, which is where the line crosses the y-axis.

• Linear functions have constant rates of change, or slope.
• The equation \(y = mx + b\) can describe any line, with coefficients \(m\) and \(b\) determining its position and steepness.
• Linear functions typically have one x-intercept and one y-intercept.

In our problem, the function is written in a different form, \(Y_1 = \frac{3x + 5}{4}\). This is essentially a linear function that can be rearranged into the standard form by multiplying everything by 4, resulting in \(4Y_1 = 3x + 5\). By setting \(Y_1 = 0\) and solving for \(x\), you find the x-intercept of this line.

Understanding Rational Equations

Rational equations are equations that involve fractions, or more specifically, fractions that have polynomials in the numerator, the denominator, or both. These types of equations often arise in real-world scenarios and require specific methods to solve because of their complex nature.

• A rational equation is in the form \(\frac{P(x)}{Q(x)} = 0\), where \(P(x)\) and \(Q(x)\) are polynomials.
• To solve a rational equation, it's often necessary to eliminate the fraction by multiplying through by the denominator.
• Care must be taken to ensure solutions do not result in division by zero.

In the original exercise, the equation \(Y_1 = \frac{3x + 5}{4}\) is a simple rational equation, where the process of solving involves removing the fraction by multiplying through by 4, yielding \(0 = 3x + 5\). This matches the step-by-step process that requires isolating \(x\) to solve for the x-intercept by working with the rational aspect of the equation.