Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-40: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ x(3 x-1) \leq 4 $$

Step-by-Step Solution

Answer

The solution is \([-1, \frac{4}{3}]\).

Step 1: Expand the Inequality

Start by expanding the left side of the inequality. The inequality is originally given as \(x(3x-1) \leq 4\). Distribute the \(x\) through the parenthesis to get \(3x^2 - x \leq 4\).

Step 2: Rearrange to Standard Quadratic Form

Rearrange the inequality into standard quadratic form \(ax^2 + bx + c \leq 0\). Subtract 4 from both sides to obtain \(3x^2 - x - 4 \leq 0\).

Step 3: Find Roots of the Quadratic Equation

To solve the inequality \(3x^2 - x - 4 = 0\), use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For this equation, \(a=3\), \(b=-1\), and \(c=-4\). Calculate the discriminant \(b^2 - 4ac = (-1)^2 - 4(3)(-4) = 1 + 48 = 49\). The square root of 49 is 7.

Step 4: Calculate the Roots

Substitute the values into the quadratic formula:\[x = \frac{-(-1) \pm 7}{2 \cdot 3}\]This simplifies to:\[x = \frac{1 \pm 7}{6}\]The two roots are \(x = \frac{8}{6} = \frac{4}{3}\) and \(x = \frac{-6}{6} = -1\).

Step 5: Test Intervals for Solutions

Since the inequality is \(3x^2 - x - 4 \leq 0\), we are looking for intervals where the parabola is non-positive. The roots divide the number line into intervals: \((-\infty, -1)\), \([-1, \frac{4}{3}]\), and \((\frac{4}{3}, \infty)\). Choose test points from each interval to determine where the inequality holds:- For \(x = -2 \in (-\infty, -1)\), \(3(-2)^2 - (-2) - 4 = 12 + 2 - 4 = 10\) (not \(\leq 0\)).- For \(x = 0 \in [-1, \frac{4}{3}]\), \(3(0)^2 - 0 - 4 = -4\) (\(\leq 0\)).- For \(x = 2 \in (\frac{4}{3}, \infty)\), \(3(2)^2 - 2 - 4 = 12 - 2 - 4 = 6\) (not \(\leq 0\)).

Step 6: Write the Solution as an Interval

Based on the test results, the solution to the inequality is within the interval \([-1, \frac{4}{3}]\) where the inequality \(3x^2 - x - 4 \leq 0\) holds true.

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. The graph of a quadratic equation is a parabola. This means it can either open upwards or downwards, depending on the sign of \(a\):

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.

Solving quadratic inequalities like \(3x^2 - x \leq 4\) involves finding where the quadratic function is less than or equal to zero. In this exercise, the first step was to expand the expression to make it easier to handle. Next, it was rearranged into the standard quadratic form so that it can be analyzed properly. Understanding these basic properties of quadratic equations is essential for solving them in different contexts, such as when they are part of an inequality.

Interval Notation

Interval notation is a convenient way to describe a range of numbers on the number line. It is especially useful in expressing the solutions of inequalities. In the context of solving quadratic inequalities, after finding the roots, we split the number line into intervals to test which satisfy the inequality.For example, the solution of the inequality \(3x^2 - x - 4 \leq 0\) in the form of interval notation is

• \([-1, \frac{4}{3}]\)

Here, the brackets \([\ ]\) indicate that the endpoints \(-1\) and \(\frac{4}{3}\) are included in the solution set, meaning the inequality holds even when \(x\) is precisely at these points. Ranges where the inequality was not valid, such as \((\frac{4}{3}, \infty)\), are not part of the solution, demonstrating how certain intervals on the number line can be the answers to inequalities.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It provides the solutions or "roots" of any quadratic equation in the form \(ax^2 + bx + c = 0\) and is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula comes in handy when the quadratic equation cannot be factored easily. It automatically produces the roots by substituting the values of \(a\), \(b\), and \(c\). In the solution provided, the quadratic formula was used to determine the roots \(x = -1\) and \(x = \frac{4}{3}\), which help in solving the inequality by identifying critical points where the function changes its sign.The term under the square root sign, \(b^2 - 4ac\), is called the discriminant:

• If it's positive, there are two distinct real roots.
• If zero, there's exactly one real root.
• If negative, the roots are complex and not real.

Understanding how and when to use the quadratic formula is a key concept in tackling inequalities involving quadratics.