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Chapter 7: Problem 370

Solve each rational inequality and write the solution in interval notation. $$\frac{5}{x-1} \leq \frac{4}{x+2}$$

Short Answer

Expert verified
The solution in interval notation is \((-\infty, -14]\).

Step by step solution

01

Find the common denominator

First, identify a common denominator for the rational expressions on both sides of the inequality. The denominators are \(x-1\) and \(x+2\). Their common denominator is \((x-1)(x+2)\).
02

Clear the denominators

Multiply both sides of the inequality by \((x-1)(x+2)\) to clear the denominators: \[ \frac{5}{x-1} (x-1)(x+2) \leq \frac{4}{x+2} (x-1)(x+2) \] This simplifies to: \[5(x+2) \leq 4(x-1) \]
03

Simplify the inequality

Distribute the terms on both sides to simplify: \[5x + 10 \leq 4x - 4 \]
04

Isolate the variable

Subtract \4x\ from both sides: \[5x - 4x + 10 \leq -4 \] This simplifies to: \[x + 10 \leq -4 \]
05

Solve for x

Subtract 10 from both sides: \[x \leq -14 \]
06

Determine the exclusion

Identify the values that make the original denominators zero, namely \(x = 1\) and \(x = -2\). These values are not included in the solution set.
07

Write the solution in interval notation

Since \(x = 1\) and \(x = -2\) are not included and we found \(x \leq -14\), the final solution set is \((-\infty, -14]\).

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Key Concepts

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

Common Denominator
When solving rational inequalities, finding a common denominator is crucial. This allows us to combine terms and simplify the inequality. In the exercise, the denominators were \(x-1\) and \(x+2\). To combine these into a single term, we multiply them together to get the common denominator:
  • \((x-1)(x+2)\)

Remember, the common denominator should be the least common multiple of the original denominators.
It's like finding a common language for two fractions. Once you have it, you can proceed to the next step.
Clearing Denominators
After finding the common denominator, the next step is to clear the denominators. This means multiplying both sides of the inequality by the common denominator.
This helps 'cancel out' the fractions, making the problem easier to work with. In our exercise, we multiplied both sides by \((x-1)(x+2)\), which turned:
  • \(\frac{5}{x-1}(x-1)(x+2) \leq \frac{4}{x+2}(x-1)(x+2)\)

Into:
  • \(5(x+2) \leq 4(x-1)\)

This 'cleared' the fractions and simplified the inequality. Be mindful to apply the same multiplication method to all terms to maintain balance.
Isolating Variables
Isolating the variable is about getting 'x' by itself on one side of the inequality. This step involves basic algebraic operations like adding, subtracting, multiplying, or dividing both sides of the inequality. In our exercise:
  • First, distribute the terms: \(5(x+2) = 5x + 10\) and \(4(x-1) = 4x - 4\).
  • Then, combine like terms: \(5x + 10 \leq 4x - 4\).
  • Subtract \(4x\) from both sides: \(x + 10 \leq -4\).
  • Finally, subtract 10 from both sides: \(x \leq -14\).

Isolating the variable helps us find the values of 'x' that satisfy the inequality.
Interval Notation
Once you solve the inequality, you express the solution in interval notation. This notation uses brackets and parentheses to show the range of values that satisfy the inequality. In the exercise, our solution was \(x \leq -14\).
We need to consider values that make the denominators zero, specifically \(x = 1\) and \(x = -2\). These values cannot be included, creating exclusions. Finally, write the solution as
  • \((-\textbackslash infty, -14] \)

Parentheses \(()\) mean the number is not included, while brackets \([]\) mean the number is included. Use interval notation for a clear and accurate representation of solution sets.

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