Question

Solve each inequality algebraically. $$6 x-5<\frac{6}{x}$$

Short answer

The solution set is \( -\frac{2}{3} < x < 0 \) or \( 0 < x < \frac{3}{2} \).

Step-by-step solution

Move all terms to one side

First, rewrite the inequality so that all terms are on one side of the inequality sign. Subtract \(\frac{6}{x}\) from both sides to get: \[6x - 5 - \frac{6}{x} < 0.\]

Common Denominator

Combine the terms by finding a common denominator, which is \(x\). This gives: \[\frac{6x^2 - 5x - 6}{x} < 0.\]

Factoring the Quadratic Expression

Factor the numerator of the rational expression. The quadratic equation \(6x^2 - 5x - 6\) factors to \((3x + 2)(2x - 3)\). Thus, the inequality becomes: \[\frac{(3x + 2)(2x - 3)}{x} < 0.\]

Roots and Test Intervals

Find the critical points by setting each factor equal to zero. This gives the roots: \(x = -\frac{2}{3}\), \(x = \frac{3}{2}\), and \(x = 0\) (since \(x = 0\) makes the denominator zero). These roots divide the number line into four intervals: \((-\infty, -\frac{2}{3})\), \((- \frac{2}{3}, 0)\), \((0, \frac{3}{2})\), and \(( \frac{3}{2}, \infty)\). Test each interval.

Testing Intervals

Choose test points from each interval to see if they make the inequality true. For \((- \frac{2}{3}, 0)\) and \((0, \frac{3}{2})\), the inequality \(\frac{(3x + 2)(2x - 3)}{x} < 0\) holds. Test points: \(-1\), \(1\), and \(2\). Check each interval for sign.

Solution Set

Combine the intervals where the inequality holds true. The solution set is \(- \frac{2}{3} < x < 0\) or \(0 < x < \frac{3}{2}\).

Key concepts

factoring quadratics

Factoring quadratics is a fundamental skill in solving inequalities. In our example, we need to factor the quadratic expression in the numerator of the rational expression to simplify the inequality. The quadratic equation is given by \(6x^2 - 5x - 6\). To factor it, we look for two binomials that multiply to give the quadratic expression.
The quadratic \(6x^2 - 5x - 6\) can be factored into \((3x + 2)(2x - 3)\).
Multiplying these two binomials out verifies the factorization:

• First: \(3x \times 2x = 6x^2\)
• Outer: \(3x \times -3 = -9x\)
• Inner: \(2 \times 2x = 4x\)
• Last: \(2 \times -3 = -6\)

This simplifies to \(6x^2 - 5x - 6\), confirming the correct factorization. Factoring helps us identify the roots, which are crucial for testing intervals and finding solutions.

test intervals

Test intervals involve finding the roots of the equation and using them to divide the number line into sections. In this problem, we find roots by setting each factor equal to zero:

• \(3x + 2 = 0 \Rightarrow x = -\frac{2}{3}\)
• \(2x - 3 = 0 \Rightarrow x = \frac{3}{2}\)

Additionally, \(x = 0\) since the denominator must not be zero. Thus, our critical points are \(-\frac{2}{3}\), \(0\), and \(\frac{3}{2}\). These critical points divide the number line into four intervals:

• \((-\infty, -\frac{2}{3})\)
• \((- \frac{2}{3}, 0)\)
• \((0, \frac{3}{2})\)
• \(( \frac{3}{2}, \infty)\)

We test each interval by choosing a number and substituting it back into the inequality. If the expression is negative in those intervals, it satisfies the inequality.

rational expressions

Rational expressions involve fractions where the numerator and the denominator are polynomials. In this inequality, we transformed the expression into a rational form: \(\frac{(3x + 2)(2x - 3)}{x} < 0\).
To solve inequalities involving rational expressions, it's important to:

• Find a common denominator when combining terms
• Simplify the expression by factoring if possible
• Identify the critical points where the numerator and the denominator are zero

Understanding rational expressions is essential because it allows us to simplify and analyze inequalities more effectively.

common denominator

Finding a common denominator is crucial for combining fractions. In this problem, we started with the inequality \(6x - 5 < \frac{6}{x}\).
To combine these terms, we moved all terms to one side, resulting in \(6x - 5 - \frac{6}{x} < 0\).
Next, we find the common denominator, which is \(x\). Rewriting the expression with this common denominator gives us:\\[\frac{6x^2 - 5x - 6}{x} < 0\].
This step makes the inequality easier to solve as combining terms into one rational expression helps to factor and find the critical points.
Remember, finding a common denominator enables you to simplify complex expressions, making them more manageable.