Question

Find the test intervals of the inequality. \(\frac{x-3}{x-1}<2\)

Short answer

The solution to the inequality is given by the intervals \( x < 1 \) and \( x > 1 \). The number \( x = 1 \) is excluded from the solution.

Step-by-step solution

- Identify the undefined points

The denominator cannot be equal to zero since division by zero is undefined in mathematics. Therefore, \( x-1 \neq 0 \), which means \( x \neq 1 \). This is the excluded value.

- Solve the inequality as an equation

Treat the inequality as an equation \( \frac{x-3}{x-1} = 2 \) to find the critical numbers. To solve the equation, cross multiply to get rid of the denominator, yielding \( x - 3 = 2x - 2 \). Solving this gives \( x = 1 \).

- Set up the test intervals

The numbers \( x = 1 \) (from step 2) and \( x = 1 \) (from step 1, the undefined point) will be used to divide the number line into intervals. These intervals will then be tested in the original inequality, to check which interval is a solution to the inequality.

- Test the intervals

The intervals are \( x < 1 \), \( x = 1 \), and \( x > 1 \). Substitute a number from each of these intervals into the original inequality \( \frac{x-3}{x-1} < 2 \). For \( x < 1 \), choose \( x = 0 \), which makes the inequality \( \frac{0-3}{0-1} < 2 \) or \( 3 > 2 \) which is true. Next, substitute \( x = 1 \), which makes the inequality \( \frac{1-3}{1-1} < 2 \) or undefined which is false. Last, substitute \( x = 2 \) as solution standing for \( x > 1 \), which makes the inequality \( \frac{2-3}{2-1} < 2 \) or \( -1 < 2 \) which is true. Therefore, the solutions to the inequality lie in the intervals \( x < 1 \) and \( x > 1 \).

Key concepts

Understanding Test Intervals

Test intervals are an essential concept when working with inequalities. They help us determine which sections of the number line satisfy an inequality.
To derive test intervals, we first need to identify crucial points that will divide the number line into different parts. These crucial points usually include:

• Critical numbers where the expression equals zero.
• Excluded values where the expression is undefined.

Once we have identified these points, we use them to partition the number line into intervals. The next step is testing each interval by picking a representative value (often referred to as a test point) from each section. We substitute these values into the original inequality to check if the inequality holds true in that interval.
This method helps confirm which sections of the number line satisfy the inequality conditions.

Exploring Critical Numbers

Critical numbers play a vital role in understanding inequalities. They are values where the expression equated to zero or to another number we are comparing, often directly relates to changes in the inequality's validity.
In the case of solving inequalities involving rational expressions, critical numbers might result from the equation derived from the inequality. For instance, using the inequality \( \frac{x-3}{x-1} < 2 \), treating it as \( \frac{x-3}{x-1} = 2 \) helps in finding such critical numbers. Solving this equation involves clearing the denominator through multiplication, simplifying to find where the numerator equals zero, such as \( x = 1 \) in our example.
Finding critical numbers is crucial as they highlight boundary points on the number line where the inequality might change its state, from true to false or vice versa. Thus, examining these points is fundamental before establishing test intervals.

Solving Inequalities: A Pathway to Solutions

Inequality solutions differ from solving equations because they involve determining ranges of values rather than single solutions. This involves several steps that lead us to these ranges and allows us to confirm them.
Starting with writing the inequality and identifying undefined values and critical numbers gives a clear direction. After finding these, the use of test intervals becomes crucial. By testing values from each interval derived from critical numbers and undefined points, we check which intervals satisfy the inequality.

• First, select test points from each interval.
• Substitute them back into the original inequality.
• Verify if the resulting statements are true or false.

Finally, the set of all values from the intervals where the inequality holds true forms the solution to the inequality. This structured approach ensures a comprehensive verification of the solution, providing a complete understanding of the inequality's solution set.