(x-1)(x-3) 0

Jasonbradley

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Jun 17, 2024

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Solving the Inequality (x-1)(x-3) > 0

This article will guide you through solving the inequality (x-1)(x-3) > 0.

Understanding the Inequality

The inequality (x-1)(x-3) > 0 asks us to find all the values of x that make the product of (x-1) and (x-3) positive.

Solving the Inequality

1. Find the critical points: The critical points are the values of x that make the expression equal to zero.

   • Set each factor equal to zero and solve:
     • x - 1 = 0 => x = 1
     • x - 3 = 0 => x = 3

2. Create a sign chart: The critical points divide the number line into three intervals:

   • x < 1
   • 1 < x < 3
   • x > 3

   Interval	x - 1	x - 3	(x-1)(x-3)
   x < 1	-	-	+
   1 < x < 3	+	-	-
   x > 3	+	+	+

   • Choose a test value within each interval and evaluate the sign of (x-1)(x-3):
     • x < 1: Choose x = 0. (0 - 1)(0 - 3) = 3 > 0
     • 1 < x < 3: Choose x = 2. (2 - 1)(2 - 3) = -1 < 0
     • x > 3: Choose x = 4. (4 - 1)(4 - 3) = 3 > 0

3. Identify the solution: The inequality (x-1)(x-3) > 0 is true when (x-1)(x-3) is positive. From our sign chart, we see this occurs in the following intervals:

   • x < 1
   • x > 3

Therefore, the solution to the inequality (x-1)(x-3) > 0 is x < 1 or x > 3.

Graphical Representation

You can also visualize the solution by plotting the graph of the function y = (x-1)(x-3). The solution represents the intervals where the graph is above the x-axis.

Key Points:

• Critical Points: Values that make the expression equal to zero.
• Sign Chart: Helps determine the sign of the expression in different intervals.
• Test Values: Used to evaluate the sign of the expression within each interval.
• Solution: The intervals where the inequality is true.