Problem 30 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ 3 x^{2}-2 x-1 \geq 0 $$

Step-by-Step Solution

Answer

The solution is \(x \in (-\infty, -\frac{1}{3}] \cup [1, \infty)\)."

Step 1: Identify the inequality

The inequality given is \(3x^2 - 2x - 1 \geq 0\). This is a quadratic inequality, which requires finding the intervals of \(x\) where the inequality holds true.

Step 2: Solve the quadratic equation

First, we solve the quadratic equation \(3x^2 - 2x - 1 = 0\) to find the critical points. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = -2\), \(c = -1\).

Step 3: Calculate the discriminant

The discriminant \(b^2 - 4ac = (-2)^2 - 4(3)(-1) = 4 + 12 = 16\). Since the discriminant is positive, there are two distinct real roots.

Step 4: Find the roots by using the quadratic formula

Substitute the values into the quadratic formula to get:\[x = \frac{-(-2) \pm \sqrt{16}}{2(3)} = \frac{2 \pm 4}{6}\].This gives the roots \(x = 1\) and \(x = -\frac{1}{3}\).

Step 5: Determine the intervals for testing

The roots divide the x-axis into intervals: \((-\infty, -\frac{1}{3})\), \((-\frac{1}{3}, 1)\), \((1, \infty)\). We need to test points from each interval in the inequality \(3x^2 - 2x - 1 \geq 0\).

Step 6: Test the intervals

Choose a test point from each interval:- For \((-\infty, -\frac{1}{3})\), pick \(-1\): \(3(-1)^2 - 2(-1) - 1 = 3 + 2 - 1 = 4 \geq 0\).- For \((-\frac{1}{3}, 1)\), pick \(0\): \(3(0)^2 - 2(0) - 1 = -1 ot\geq 0\).- For \((1, \infty)\), pick \(2\): \(3(2)^2 - 2(2) - 1 = 12 - 4 - 1 = 7 \geq 0\).

Step 7: Include the roots in the solution set

Since the original inequality is \(\geq\), the roots \(x = 1\) and \(x = -\frac{1}{3}\) also satisfy the inequality because substituting these values into the inequality results in zero, which is valid.

Step 8: Write the solution

The solution set includes the intervals where the inequality holds and the roots: \(x \in (-\infty, -\frac{1}{3}] \cup [1, \infty)\).

Key Concepts

Quadratic Formula

When solving quadratic inequalities, the quadratic formula is a powerful tool we use to find the roots of the related quadratic equation. The formula is given as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's a breakdown of the components:

• 🅰️ **`a`**: This is the coefficient of \(x^2\), representing the leading factor of the equation.
• 🅱️ **`b`**: This is the coefficient of \(x\), the linear component.
• 🚫 **`c`**: This is the constant term, the standalone number in the equation.

To use the quadratic formula, substitute the values of \(a\), \(b\), and \(c\) from your quadratic equation. This helps you find where the curve intercepts the x-axis, which is critical to understanding the solution of inequalities.
Remember, these intercepts are also known as the "critical points," where the sign of the inequality may change.

Discriminant

The discriminant is an essential part of determining the nature of the roots of a quadratic equation. It is the part under the square root in the quadratic formula:\[ b^2 - 4ac \]This single value can tell you several things about the solutions of the equation:

• ♾️ **Positive Discriminant:** If \(b^2 - 4ac > 0\), there are two real and distinct roots. It means the parabola intersects the x-axis at two points.
• 📏 **Zero Discriminant:** If \(b^2 - 4ac = 0\), there is exactly one real root. The parabola touches the x-axis at just one point, which is a double root.
• 🔒 **Negative Discriminant:** If \(b^2 - 4ac < 0\), there are no real roots; instead, the roots are complex. Here, the parabola does not intersect the x-axis at all.

In our example, a positive discriminant of 16 indicated the presence of two distinct real roots which allowed us to test intervals determined by these roots to solve the inequality.

Real Roots

Real roots are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These are points where the graph of the quadratic equation, a parabola, will intersect the x-axis.
Significance of real roots in inequalities:

• 🔗 **Root Locations:** They divide the x-axis into intervals which must be tested to determine where the inequality holds true.
• ✅ **Inclusion of Roots:** In inequalities with "greater than or equal to" or "less than or equal to," the roots are included in the solution set since they equate to zero, fitting the equality condition.

During interval testing, you pick test values from each segment created by the roots and check which intervals satisfy the inequality.
This ensures that you don't miss any hidden solutions, fully covering all valid possibilities for \(x\).