Problem 50 - A Graphical Approach to College Algebra

Question

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \((x-3)^{2 / 5}=(4 x)^{1 / 5}\) (b) \((x-3)^{2 / 5}>(4 x)^{1 / 5}\) (c) \((x-3)^{2 / 5}<(4 x)^{1 / 5}\)

Step-by-Step Solution

Answer

(a) Solutions are \(x = 1\) and \(x = 9\). (b) Solution is \(x < 1\). (c) Solution is \(1 < x < 9\).

Step 1: Solve the Equation

We start by solving the equation \((x-3)^{2/5} = (4x)^{1/5}\). To eliminate the fractional exponents, raise both sides to the power of 5, resulting in \((x-3)^2 = 4x\).

Step 2: Simplify the Resulting Polynomial

Expand the left side to get \(x^2 - 6x + 9 = 4x\). Then, bring all terms to one side to form a quadratic equation: \(x^2 - 10x + 9 = 0\).

Step 3: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -10\), \(c = 9\). Plug these values into the formula: \(x = \frac{10 \pm \sqrt{(-10)^2 - 4 \times 1 \times 9}}{2 \times 1}\). This simplifies to \(x = \frac{10 \pm \sqrt{64}}{2}\).

Step 4: Calculate the Roots

Calculate the values by solving \(x = \frac{10 \pm 8}{2}\). Thus, the solutions are \(x = 9\) and \(x = 1\).

Step 5: Verify Solutions Graphically

Graph both \(y = (x-3)^{2/5}\) and \(y = (4x)^{1/5}\). Verify that the graphs intersect at \(x = 9\) and \(x = 1\).

Step 6: Solve the Inequality for (b)

For \((x-3)^{2/5} > (4x)^{1/5}\), evaluate the graph regions where the curve \((x-3)^{2/5}\) is above \((4x)^{1/5}\). The inequality holds in the region \(x < 1\).

Step 7: Solve the Inequality for (c)

For \((x-3)^{2/5} < (4x)^{1/5}\), evaluate where the curve \((x-3)^{2/5}\) is below \((4x)^{1/5}\). This inequality holds in the region \(1 < x < 9\).

Key Concepts

Fractional Exponents

Fractional exponents might seem tricky at first, but they are quite approachable once you understand their purpose. Simply put, a fractional exponent is another way to express a root. For example, when you see something like \((x - 3)^{2/5}\), it means the 5th root of \((x-3)\) is being squared.

• The base in the expression, which is \((x-3)\), is the number we're working with.
• The numerator of the fraction (here, 2) is the power to which the base is raised.
• The denominator (5 in this case) signifies the root you need to calculate.

To eliminate fractional exponents, you can raise both sides of the equation to the power of the denominator, simplifying the expression into a more traditional form. In our exercise, raising both sides to the 5th power transforms fractional exponents into integers, paving the way for familiar algebraic manipulation. This step reveals our quadratic equation, which is easier to solve.

Quadratic Equations

Quadratic equations are foundational in algebra, defined generally as equations of the form \(ax^2 + bx + c = 0\).They are solved by factoring, completing the square, or employing the quadratic formula.

• In our example, the equation \((x-3)^2 = 4x\) transforms into \(x^2 - 10x + 9 = 0\) after simplification.
• The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) can be used here, where in our equation, \(a = 1\), \(b = -10\), \(c = 9\).

Using the quadratic formula provides us with the roots of the equation. The discriminant, \(b^2 - 4ac\), indicates the nature of the roots:

• If positive, there are two distinct real roots.
• If zero, one real root exists.
• If negative, the roots are complex.

Here, we find two real roots: \(x = 9\) and \(x = 1\). Understanding quadratics helps uncover these solutions, which can then be verified through a graphical representation.

Graphical Solutions

Graphical solutions provide a visual representation of equations and inequalities, making them easier to comprehend. By plotting the functions \(y = (x-3)^{2/5}\) and \(y = (4x)^{1/5}\), we can see where these curves intersect or diverge.

• Intersecting points at \(x = 9\) and \(x = 1\) are the solutions to our original equation.
• These graphs help us understand the behavior of functions beyond numerical solutions.

For inequalities, the visualization shows where one function lies above or below another. For instance, to solve \((x-3)^{2/5} > (4x)^{1/5}\), we look for regions where the curve of \((x-3)^{2/5}\) lies above \((4x)^{1/5}\).Graphical methods are great because they unveil a broader picture of the relationships we are studying. This enhances our understanding and solidifies the solutions obtained algebraically.

Inequalities in Algebra

Inequalities in algebra express a relationship where one quantity is greater or less than another; they can be strict or inclusive. Understanding how to solve these through both algebraic manipulation and graphical interpretation is crucial.

• Given the inequality \((x-3)^{2/5} > (4x)^{1/5}\), the solution involves identifying where one curve surpasses the other.
• A graphical method highlights that the inequality holds where \(x < 1\).
• For \((x-3)^{2/5} < (4x)^{1/5}\), the solution is apparent in the interval \(1 < x < 9\).

Inequalities demand attention to regions on the graph rather than single points. This means interpreting where the functions graphically diverge or converge, providing a real-world context as these solutions often represent ranges of possible values. Approaching inequalities with both algebraic techniques and graphical insights ensures a comprehensive solution to these problems.