Solving Quadratic Inequalities: Understanding 3x1 ≥ 2x3

When dealing with mathematical inequalities, one common problem is to determine when one expression is greater than or equal to another. In this article, we will explore the inequality 3x1 ≥ 2x3 and provide a detailed solution using algebraic methods.

Introduction to the Concept

Understanding whether 3x1 ≥ 2x3 is not a straightforward task for all values of x. This requires a thorough analysis using mathematical principles and algebraic techniques. This article will guide you through the steps to solve this inequality and help you understand the underlying mathematical concepts.

Algebraic Analysis of the Inequality

Let's start by applying the square root property to the inequality 3x1 ≥ 2x3. By squaring both sides, we get:

Squaring Both Sides

Starting from the inequality:

[ 3x1 ≥ 2x3 ]

We square both sides:

[ (3x1)^2 ≥ (2x3)^2 ]

This leads to:

[ 9x^2 - 6x - 8 ≥ 0 ]

Solving the Quadratic Inequality

To solve the inequality 9x^2 - 6x - 8 ≥ 0, we first find the roots of the corresponding quadratic equation:

[ 5x^2 - 6x - 8 0 ]

Solving this quadratic equation using the quadratic formula:

[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

Substituting the coefficients a 5, b -6, and c -8:

[ x frac{-(-6) pm sqrt{(-6)^2 - 4 cdot 5 cdot (-8)}}{2 cdot 5} ]

This simplifies to:

[ x frac{6 pm sqrt{36 160}}{10} ]

Further simplification gives:

[ x frac{6 pm sqrt{196}}{10} ]

Therefore:

[ x frac{6 pm 14}{10} ]

This results in two solutions:

[ x frac{20}{10} 2 ] and [ x frac{-8}{10} -frac{4}{5} ]

Interval Analysis

Now, we need to determine the intervals where the inequality 9x^2 - 6x - 8 ≥ 0 holds true. The intervals are:

x -frac{4}{5} -frac{4}{5} ≤ x ≤ 2 x 2

Next, we need to test these intervals to see where the inequality is satisfied.

Testing Intervals

Choose a test point from each interval and substitute it back into the inequality:

x -1: (Test point for x -frac{4}{5}):

Substitute x -1 into the inequality:

[ 5(-1)^2 - 6(-1) - 8 5 6 - 8 3 ≥ 0 ]

This satisfies the inequality, so x -frac{4}{5} is part of the solution set.

x 0: (Test point for -frac{4}{5} ≤ x ≤ 2):

Substitute x 0 into the inequality:

[ 5(0)^2 - 6(0) - 8 -8 0 ]

This does not satisfy the inequality, so -frac{4}{5} ≤ x ≤ 2 is not a part of the solution set.

x 3: (Test point for x 2):

Substitute x 3 into the inequality:

[ 5(3)^2 - 6(3) - 8 45 - 18 - 8 19 ≥ 0 ]

This satisfies the inequality, so x 2 is part of the solution set.

Conclusion

Therefore, the inequality 3x1 ≥ 2x3 holds true when:

x ≤ -frac{4}{5} x ≥ 2

This result can be summarized as:

[ 3x1 ≥ 2x3 text{ when } x ≤ -frac{4}{5} text{ or } x ≥ 2 ]

Understanding and solving such inequalities is a crucial skill in algebra and mathematics, providing a deeper insight into the behavior of functions and their relationships.