Problem 21 - Algebra A Combined Function

Question

Solve. Write the solution set in interval notation. $$ \frac{3}{x-2}<4 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, 2) \cup \left(\frac{11}{4}, \infty\right)\).

Step 1: Understand the Inequality

The inequality given is \( \frac{3}{x-2} < 4 \). We need to find the values of \( x \) that satisfy this inequality. Keep in mind that \( x eq 2 \) because the denominator cannot be zero.

Step 2: Isolate the Variable

To remove the fraction, we'll multiply both sides of the inequality by \( x - 2 \). However, we need to consider two cases because multiplying by a negative number reverses the inequality sign.

Step 3: Case 1: \( x-2 > 0 \)

For \( x - 2 > 0 \), \( x > 2 \). Multiply both sides by \( x-2 \): \[ 3 < 4(x-2) \]Simplify:\[ 3 < 4x - 8 \]Add 8 to both sides:\[ 11 < 4x \]Divide by 4:\[ \frac{11}{4} < x \]Thus the solution for this case is \( x > \frac{11}{4} \).

Step 4: Case 2: \( x-2 < 0 \)

For \( x - 2 < 0 \), \( x < 2 \). Multiply both sides by \( x-2 \): \[ 3 > 4(x-2) \]Simplify:\[ 3 > 4x - 8 \]Add 8 to both sides:\[ 11 > 4x \]Divide by 4:\[ \frac{11}{4} > x \]Thus the solution for this case is \( x < \frac{11}{4} \) and \( x < 2 \).

Step 5: Combine the Solutions

The solution for \( x > \frac{11}{4} \) is valid for \( x > 2 \), and the solution for \( x < \frac{11}{4} \) must also be \( x < 2 \). The combined solution is when \( x < \frac{11}{4} \) or \( x > 2 \). Remove values where \( x = 2 \). So, the interval notation is:\[(-\infty, 2) \cup \left(\frac{11}{4}, \infty\right)\]

Step 6: Check the Solution

Verify the solution by choosing values within the intervals, such as 1 for \( x < 2 \) and 3 for \( x > \frac{11}{4} \). Substitute in the original inequality to ensure they satisfy \( \frac{3}{x-2} < 4 \). Both choices satisfy the inequality, confirming our solution is correct.

Key Concepts

Interval Notation

Interval notation is a way of expressing solution sets for inequalities. It allows us to show which parts of the number line are included in the solution without using words.
For instance, if we have a range of numbers, we can describe them with brackets:

• Round parentheses, like \((a, b)\), indicate that the endpoints \(a\) and \(b\) are not included in the interval.
• Square brackets, like \([a, b]\), mean the endpoints are included in the interval.

In the solution to the inequality \(\frac{3}{x-2} < 4\), we found two separate intervals indicated by the inequality solutions. The final result was written as \((-\infty, 2) \cup \left(\frac{11}{4}, \infty\right)\).
This notation means:

• For \((-\infty, 2)\), all numbers less than 2 are part of the solution, but 2 is not included (hence the parenthesis).
• The symbol \(\cup\) represents the union of sets, indicating that either solution set will satisfy the inequality.

Rational Inequalities

Rational inequalities are inequalities that involve a fraction where the numerator and/or the denominator is a polynomial. These inequalities can be tricky because they involve careful consideration of when the fraction changes sign, especially around points that make the denominator zero.

In equations like \(\frac{3}{x-2} < 4\), the main challenge is managing the critical point, in this case, \(x=2\), which would make the denominator zero and thus undefined.

To solve a rational inequality:

• Identify values that would make the denominator zero, and exclude them from your solution.
• Consider different cases for the sign of the expression, typically around these critical points.
• Analyze each case separately to see how the inequality behaves on either side of these critical points.

Inequality Solutions

Solving inequalities involves finding all the possible values of a variable that make the inequality true. This usually requires manipulating the inequality in similar ways to solving equations, but with extra care when multiplying or dividing by negative numbers, because this reverses the inequality sign.

For the inequality \(\frac{3}{x-2}<4\), we broke it into two cases based on: \(x-2>0\) and \(x-2<0\). Solving these separately ensures each section of the inequality is properly considered.

Remember these key steps when solving inequalities:

• Isolate the variable on one side of the inequality wherever possible.
  
• Pay attention to the sign of expressions particularly if they flip the inequality.
• Combine solutions, using logical operators like "or" to capture all solutions to the original inequality.

Finally, checking your solutions using test values within your intervals ensures accuracy. This confirmation step helps guarantee that the inequality holds across the solution set.