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Chapter 2: Problem 97

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I began the solution of the rational inequality \(\frac{x+1}{x+3} \geq 2\) by setting both \(x+1\) and \(x+3\) equal to zero.

Short Answer

Expert verified
The statement doesn't make sense. The right way to solve the inequality is to rearrange it to have 0 at one side, simplify to a fraction, identify critical points and then check the intervals to identify where the inequality holds true.

Step by step solution

01

Identify the Mistake

The statement can be regarded as incorrect because it's not valid to set \(x+1\) and \(x+3\) to zero in an attempt to solve the inequality \(\frac{x+1}{x+3} \geq 2\). Instead, we have to solve this inequality by isolating x.
02

Correct Approach

The correct way to do this is to first subtract 2 from both sides to get the equation to the form 0 on the RHS. \(\frac{x+1}{x+3} - 2 \geq 0\) and then get common denominator for both fractions.
03

Simplify the equation

After simplifying, we have simplifiable formula of \(\frac{-x-5}{x+3} \geq 0\), from which we can pick the roots.
04

Check the intervals

Now we split the interval into the three segments, which are divided by the roots -3 and -5. In each segment, we take a number and place it in simplified inequality and check the sign. This will tell where the inequality is satisfied and gives the solution.

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Key Concepts

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

Inequality Solving
When dealing with inequalities, your goal is to find the range of values for which the inequality holds true. Unlike solving a regular equation, inequalities will have a range of values rather than a distinct solution. To solve an inequality:
  • Begin by manipulating the inequality into a simpler form. Rearrange it so one side is zero, making assessment easier.
  • In our original problem, the inequality is rewritten as \(\frac{x+1}{x+3} - 2 \geq 0\).
  • Next, manipulate this inequality to a common denominator to simplify.
It's important to handle inequalities delicately because multiplying or dividing by a negative number flips the inequality sign. Always check for such operations explicitly, so you don鈥檛 accidentally change the inequality direction. Inequality solutions often entail finding critical points or roots and testing intervals based on these.
Rational Expressions
Understanding rational expressions is crucial for navigating through rational inequalities. A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. Here, the fraction \(\frac{x+1}{x+3}\) in our problem is an example of a rational expression.
  • Rational expressions can often be inconsistent at points where the denominator equals zero since division by zero is undefined.
  • In our inequality problem, setting the denominator \(x+3 = 0\) reveals a potential problem point at \(x = -3\). This is a restriction to keep in mind.
Dealing with rational expressions, particularly in inequalities, requires careful management of these undefined points. Use these as anchors in your interval testing to define where the rational expression is invalid or undefined. This is often a critical step in rational inequality solving.
Interval Testing
Interval testing involves checking different segments of the real number line against an inequality to establish where the inequality holds true. After simplifying your rational inequality and finding its roots, split the number line at these points and conduct interval testing.
  • In our example, roots were found at \(-3\) and \(-5\) from the simplified inequality \(\frac{-x-5}{x+3} \geq 0\).
  • We divide the real line into intervals based on these roots: \((-\infty, -5)\), \((-5, -3)\), and \((-3, \infty)\).
  • Pick a sample value in each interval, substitute it back into the simplified inequality, and check if the expression is positive, negative, or zero.
The signs you find for each interval help determine where the original inequality holds true. This process is crucial since it helps you see the behavior of the rational function across different ranges, empowering you to solve inequalities confidently.

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