Problem 52 - Intermediate Algebra

Question

Solve each inequality. $$ \frac{x-1}{x-2}<-1 $$

Step-by-Step Solution

Answer

The solution is \(\frac{3}{2} < x < 2\).

Step 1: Rearrange the inequality

First, we will rearrange the inequality to make it easier to solve. Start by adding 1 to both sides:\[\frac{x-1}{x-2} + 1 < 0\]

Step 2: Combine fractions

We can combine the terms on the left-hand side into a single fraction:\[\frac{x-1}{x-2} + \frac{x-2}{x-2} < 0\]This becomes:\[\frac{x-1 + (x-2)}{x-2} < 0\]

Step 3: Simplify the numerator

Simplify the numerator by combining like terms:\[\frac{2x-3}{x-2} < 0\]

Step 4: Determine critical points

The critical points are where the inequality changes sign (numerator and denominator are zero). Set the numerator equal to zero:\[2x-3 = 0 \]Solve for \(x\):\[x = \frac{3}{2}\]Also set the denominator equal to zero:\[x-2 = 0 \]Solve for \(x\):\[x = 2\]

Step 5: Test intervals

We have critical points at \(x = \frac{3}{2}\) and \(x = 2\). Test intervals:- For \(x < \frac{3}{2}\), let \(x = 1\): \( \frac{2(1)-3}{1-2} = \frac{-1}{-1} = 1 > 0 \)- For \(\frac{3}{2} < x < 2\), let \(x = \frac{3}{2} + 0.1 = 1.6\): \( \frac{2(1.6)-3}{1.6-2} = \frac{0.2}{-0.4} = -0.5 < 0 \)- For \(x > 2\), let \(x = 3\): \( \frac{2(3)-3}{3-2} = \frac{3}{1} = 3 > 0 \)

Step 6: Write the solution

The interval where the inequality \(\frac{2x-3}{x-2} < 0\) is true is when \(\frac{3}{2} < x < 2\).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions with fractions where the numerator and the denominator are algebraic expressions. These types of inequalities are solved by finding where the expression is greater than, less than, or equal to a certain value.
For example, the inequality \( \frac{x-1}{x-2} < -1 \) involves a rational expression. To solve it, we first manipulate it to a more manageable form, often by getting all terms on one side of the inequality.
These inequalities can often have critical points where the expression is undefined or equals zero, as the sign of the inequality can change at these points. It is important to identify and test these intervals to find where the inequality is satisfied.
The solution often involves a combination of interval testing and understanding the behavior of the rational expression at different points.

Critical Points

Critical points occur where numerator or denominator equals zero, a fundamental aspect in solving rational inequalities. In the expression \( \frac{2x-3}{x-2} < 0 \), critical points arise from settings where either \(2x-3=0\) or \(x-2=0\).

• For \(2x-3=0\), solving gives \(x=\frac{3}{2}\). This is where the numerator is zero.
• For \(x-2=0\), solving gives \(x=2\). This is where the denominator is zero, indicating a vertical asymptote.

These points are crucial, as they help divide the number line into intervals for testing. At these points, the rational expression might change signs, leading to different solutions in distinct intervals. Understanding critical points helps in not only solving inequalities but also in graphing rational functions and determining behavior near zeros and undefined values.

Interval Testing

Once critical points are identified, interval testing follows to determine where the inequality holds true. The process involves selecting test points from each interval divided by critical points.
For the inequality \(\frac{2x-3}{x-2} < 0\), critical points are \(x = \frac{3}{2}\) and \(x = 2\). These points divide the number line into three intervals: \(x < \frac{3}{2}\), \(\frac{3}{2} < x < 2\), and \(x > 2\).

• For \(x < \frac{3}{2}\), testing a point like \(x = 1\) shows the inequality is \(> 0\), not satisfied.
• For \(\frac{3}{2} < x < 2\), testing \(x = 1.6\) shows \(< 0\), which is satisfied.
• For \(x > 2\), testing \(x = 3\) shows \(> 0\), not satisfied.

Interval testing allows you to confidently determine where on the number line the inequality expression maintains its condition, thus providing the solution set.

Algebraic Expressions

Algebraic expressions form the foundation of rational inequalities. In these problems, expressions often consist of variables, constants, and arithmetic operations structured in terms like \(\frac{2x-3}{x-2}\).
These expressions need to be manipulated using algebraic techniques to simplify or rearrange them. Simplifying usually involves combining like terms or factoring to more clearly identify numerical relationships between variables.
If you have the inequality \( \frac{x-1}{x-2} < -1 \), combining fractions and simplifying results in \( \frac{2x-3}{x-2} < 0 \). This is a crucial step that can make solving easier.
Understanding how to work with algebraic expressions helps solve complex inequalities and further prepares students for more advanced topics in algebra and calculus.