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Chapter 2: Problem 47

Solving a Rational Inequality In Exercises \(39-52\) , solve the inequality. Then graph the solution set. $$\frac{1}{x-3} \leq \frac{9}{4 x+3}$$

Short Answer

Expert verified
The solution to the inequality is \(x \geq 6\) with the exceptions where \(x \neq 3\) and \(x \neq -\frac{3}{4}\]

Step by step solution

01

Subtracting \(9/(4x+3)\) from both sides

Subtract \(9/(4x+3)\) from both sides of the inequality to have all terms on one side and zero on the other. This gives: \(\frac{1}{x-3} - \frac{9}{4x+3} \leq 0\)
02

Finding the Least Common Denominator (LCD)

The Least Common Denominator (LCD) of \(x-3\) and \(4x+3\) is \((x-3)(4x+3)\). We multiply each term by the LCD to clear the fraction giving: \[\frac{(x-3)(4x+3)}{x-3} - \frac{(x-3)(4x+3)}{4x+3} \leq 0\]
03

Simplifying the inequality

The (x-3) terms and the (4x+3) terms cancel out respectively, resulting in the inequality: \[4x+3 - 9x + 27 \leq 0\]
04

Arrange in standard form

Rearrange and simplify the inequality in the standard quadratic form, which results in \[-5x + 30 \leq 0\].
05

Solve for x

Divide through by -5 to isolate x yielding: \[x - 6 \geq 0 \] Note that the inequality symbol flipped because we divided by a negative number. Solving for x gives: \[x \geq 6\].
06

Check the domain

We have to check the domain of the original inequality. For \(1/(x-3)\), x should not be equal to 3 because it would make the denominator zero. Same applies for \(9/(4x+3)\), x should not be -3/4. So we modify our solution to \[x \geq 6\], \(x \neq 3\] and \( x \neq -\frac{3}{4}\]\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Least Common Denominator
When solving rational inequalities, the Least Common Denominator (LCD) is essential. Here, the rational inequality involves the expressions \(\frac{1}{x-3}\) and \(\frac{9}{4x+3}\). To eliminate the fractions, we multiply each term by the LCD. The LCD of denominators \(x-3\) and \(4x+3\) is the product \((x-3)(4x+3)\). This enables us to convert the inequality into a more manageable form without fractions. It's important to be precise here to ensure the inequality holds. If errors occur in determining the LCD, the entire inequality solution may be incorrect.
Simplifying Inequalities
After applying the Least Common Denominator, the inequality becomes easier to handle. We simplify by canceling the common factors in the numerator and denominator. Subtracting \[\frac{9}{4x+3}\] from \[\frac{1}{x-3}\] leads to the expression: \[4x + 3 - 9x + 27 \leq 0\].The terms are combined and rearranged to make the inequality simpler. The goal of simplifying is to express the inequality in a form where one can solve for \(x\). Remember, simplification errors can lead to wrong conclusions, so check each step carefully.
Domain Restrictions
Domain restrictions are crucial when dealing with rational inequalities. They ensure solutions don't include values that make the denominators zero. For the original inequality:
  • \(\frac{1}{x-3}\) makes \(x eq 3\)
  • \(\frac{9}{4x+3}\) makes \(x eq -\frac{3}{4}\)
When you solve the inequality and arrive at \(x \geq 6\), you must recheck these domain restrictions and exclude these values from the solution set. These seem like minor details, but neglecting domain restrictions can lead to including invalid solutions.
Graphing Solutions
Graphing helps visualize where a rational inequality holds true. Once the solution \(x \geq 6\) is established, you'll represent this on a number line or coordinate plane. Consider the domain restrictions and ensure values \(x = 3\) and \(x = -\frac{3}{4}\) are not part of the solution.
  • Use an open circle to indicate values excluded from the solution set, such as at \(x = 3\) and \(x = -\frac{3}{4}\).
  • A shaded area shows where the inequality is satisfied, here starting from \(x = 6\).
Graphing provides a quick reference to see the solution's range and limitations.

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