Problem 56 - A Graphical Approach to Precalculus with Limits

Question

Use analytic or graphical methods to solve the inequality. $$\sqrt[3]{4 x^{2}+3}<\sqrt[3]{3-x}$$

Step-by-Step Solution

Answer

The solution to the inequality is \( -\frac{1}{4} < x < 0 \).

Step 1: Understand the Inequality

The given inequality is \( \sqrt[3]{4x^2 + 3} < \sqrt[3]{3 - x} \). Our goal is to find the values of \( x \) that satisfy this inequality.

Step 2: Remove Cube Roots

To simplify the inequality, remove the cube roots by cubing both sides of the inequality. This gives us:\[ 4x^2 + 3 < 3 - x \].

Step 3: Rearrange the Equation

Rearrange the inequality to bring all terms involving \( x \) to one side:\[ 4x^2 + x + 3 < 3 \].

Step 4: Simplify the Inequality

Subtract 3 from both sides to simplify:\[ 4x^2 + x < 0 \].

Step 5: Factor the Quadratic Expression

Factor out the common term in the inequality:\[ x(4x + 1) < 0 \].

Step 6: Solve the Factored Inequality

Determine the values of \( x \) that make \( x(4x + 1) < 0 \). This occurs between the roots of the equation when \( x = 0 \) and \( 4x + 1 = 0 \), or when \( x = -\frac{1}{4} \). The intervals to consider are: \( x < 0 \) and \( x > -\frac{1}{4} \).

Step 7: Identify the Interval

Since \( x(4x + 1) < 0 \), the solution is in the interval where \( -\frac{1}{4} < x < 0 \).

Key Concepts

Cube Root Inequalities

Inequalities involve finding the range of values for variables that make an equation true. Cube root inequalities, like the one we are solving, incorporate cube roots in the expression.

• The cube root, denoted as \( \sqrt[3]{x} \), is the value that needs to be multiplied by itself three times to give \( x \).
• To solve cube root inequalities analytically, you typically eliminate the cube roots by cubing both sides of the inequality.

Cubing both sides is valid because raising to an odd power maintains the inequality direction.
Cubed inequalities help simplify the equation into a polynomial, which clarifies the steps needed to identify solution sets. The values of \( x \) making the inequality true are found through steps such as combining like terms and factoring.

Quadratic Inequalities

Quadratic inequalities involve expressions of the form \( ax^2 + bx + c < 0 \) or \( > 0 \). Solving them means finding values of \( x \) for which the inequality holds true.

• After simplifying the inequality, it often forms a quadratic equation.
• These inequalities can have two possible solution forms: values between the roots or outside the roots.

To tackle them, first bring all terms to one side of the inequality.
Next, convert the expression into standard quadratic form.
This will help in determining the critical points, where the expression equals zero. These points split the \( x \)-axis into intervals that can be tested to see where the inequality holds.

Factoring Techniques

Factoring is a crucial step in solving polynomial inequalities. It involves writing a polynomial as the product of its factors.

• In our exercise, after removing cube roots and simplifying, the inequality was converted into \( 4x^2 + x < 0 \).
• This expression can be factored by taking out a common factor, which in our case was \( x \), resulting in \( x(4x + 1) < 0 \).

The different techniques to factor a quadratic expression include:

• Taking out common factors: As shown, this step simplifies the equation, making it easier to solve.
• Splitting middle term: When feasible, this helps break down more complex expressions.

After factoring, identify where the product of factors is less than zero by examining the intervals between and around the roots.