Problem 48 - A Graphical Approach to College Algebra

Question

Solve each equation and inequality, where \(k\) is a positive constant. (a) \(4 x^{3}-k x=0\) (b) \(4 x^{3}-k x>0\)

Step-by-Step Solution

Answer

(a) \( x = 0, \pm\frac{\sqrt{k}}{2}\). (b) \( x \in (-\infty, -\frac{\sqrt{k}}{2}) \cup (\frac{\sqrt{k}}{2}, \infty) \).

Step 1: Factor the Equation for Part (a)

For the equation \[ 4x^3 - kx = 0, \] start by factoring out the common term \( x \):\[ x(4x^2 - k) = 0. \] This gives two possible solutions: either \( x = 0 \) or \( 4x^2 - k = 0 \).

Step 2: Solve the Quadratic Expression for Part (a)

Set the quadratic expression equal to zero and solve for \( x \):\[ 4x^2 - k = 0 \quad \Rightarrow \quad 4x^2 = k \quad \Rightarrow \quad x^2 = \frac{k}{4}. \]Take the square root on both sides to find:\[ x = \pm \frac{\sqrt{k}}{2}. \] Thus, the solutions to the equation are \( x = 0, \frac{\sqrt{k}}{2}, \) and \(-\frac{\sqrt{k}}{2}.\)

Step 3: Interpret the Solution for Part (b)

For the inequality \[ 4x^3 - kx > 0, \] we start from the factorized form:\[ x(4x^2 - k) > 0. \] This inequality implies that either both terms are positive or both are negative.

Step 4: Test Intervals for Part (b)

Determine the intervals created by the critical points \( x = 0, x = \frac{\sqrt{k}}{2}, \) and \( x = -\frac{\sqrt{k}}{2} \). Test intervals: 1. \( (-\infty, -\frac{\sqrt{k}}{2}) \): Both \( x < 0 \) and \( 4x^2 - k < 0 \). 2. \( (-\frac{\sqrt{k}}{2}, 0) \): \( x < 0 \) while \( 4x^2 - k > 0 \).3. \( (0, \frac{\sqrt{k}}{2}) \): Both \( x > 0 \) and \( 4x^2 - k < 0 \) fails. 4. \( \left( \frac{\sqrt{k}}{2}, \infty \right) \): Both \( x > 0 \) and \( 4x^2 - k > 0 \).Based on testing, the inequality is satisfied in the intervals \( (-\infty, -\frac{\sqrt{k}}{2}) \cup (\frac{\sqrt{k}}{2}, \infty) \).

Step 5: Solution Summary

For part (a), the solutions are \( x = 0, \frac{\sqrt{k}}{2}, \) and \(-\frac{\sqrt{k}}{2}.\) For part (b), the solution in interval form is \[ x \in (-\infty, -\frac{\sqrt{k}}{2}) \cup (\frac{\sqrt{k}}{2}, \infty). \]

Key Concepts

Inequalities

Inequalities are mathematical expressions used to compare two values or expressions. They show the relationship between quantities that are not equal. For example, the inequality \(4x^3 - kx > 0\) tells us that the expression on the left is greater than zero.

There are different types of inequalities, such as:

• Strict inequalities like \(a > b\) or \(a < b\), where \(a\) is either greater than or less than \(b\), respectively.
• Non-strict inequalities like \(a \geq b\) or \(a \leq b\), where \(a\) is either greater or equal to, or less or equal to \(b\).

In solving inequalities, we often need to find the range of values for which the inequality holds true. This often involves interval testing, where specific intervals determined by critical points are checked to find valid solutions. Always take care when multiplying or dividing inequalities by a negative number, as the inequality sign reverses in such cases.

Factoring

Factoring is a process of breaking down an expression into simpler components, often called factors, that when multiplied together produce the original expression. In our example, the equation \(4x^3 - kx = 0\) is factored as \(x(4x^2 - k) = 0\).

To factor an expression, look for common terms or apply special factoring rules, such as:

• Factoring out the Greatest Common Factor (GCF).
• Using special patterns, like the difference of squares \((a^2 - b^2 = (a+b)(a-b))\).
• Grouping terms in quadratic expressions to simplify them.

Factoring is a crucial step in solving quadratic equations and inequalities, as it simplifies the process and reveals possible solutions or critical points.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically having the form \(ax^2 + bx + c\). In our exercise, we encounter the expression \(4x^2 - k\).

Solving quadratic equations often involves using techniques such as:

• Factoring, when the expression can be easily broken down into simpler factors.
• The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), when factoring is complex or not possible.
• Completing the square to transform the quadratic into a perfect square trinomial.

Quadratics can have real or complex roots, depending on the value of the discriminant \(b^2 - 4ac\). In our context, the quadratic expression \(4x^2 - k\) is solved by setting it to zero, giving possible solutions for \(x = \pm \frac{\sqrt{k}}{2}\).

Interval Testing

Interval testing, also known as interval analysis, is a method used to determine where certain conditions are true for expressions or inequalities. For the inequality \(x(4x^2 - k) > 0\), interval testing helps find where this condition holds.

This involves:

• Identifying critical points, which are values of \(x\) making expression terms zero, such as \(x=0\) and \(x=\pm \frac{\sqrt{k}}{2}\).
• Dividing the number line into intervals between these points.
• Testing a sample point from each interval to see if it satisfies the inequality.

After interval testing, the intervals where the inequality holds are chosen, like \((-\infty, -\frac{\sqrt{k}}{2})\) and \((\frac{\sqrt{k}}{2}, \infty)\) in our case. This step is essential, especially when expressions cannot be easily visualized or graphed, allowing an analytical approach to determine solution sets.