What is the solution to the inequality $x(x-3)>0$?
A) $x \leq 0$ or $x \geq 3$
B) $0
C) $x<0$ or $x>3$
D) $x \leq 0$ and $x \geq 3$
Answer (1)
Find the critical points by setting x ( x − 3 ) = 0 , which gives x = 0 and x = 3 .
Test the intervals ( − ∞ , 0 ) , ( 0 , 3 ) , and ( 3 , ∞ ) to determine where the inequality holds.
The inequality 0"> x ( x − 3 ) > 0 holds for x < 0 and 3"> x > 3 .
The solution to the inequality is x < 0 or 3"> x > 3 , so the answer is 3}"> x < 0 or x > 3 .
Explanation
Understanding the Inequality We are given the inequality 0"> x ( x − 3 ) > 0 . To solve this inequality, we first find the critical points by setting x ( x − 3 ) = 0 .
Finding Critical Points The critical points are the values of x that make the expression equal to zero. So, we have x = 0 or x − 3 = 0 , which means x = 0 or x = 3 . These critical points divide the number line into three intervals: ( − ∞ , 0 ) , ( 0 , 3 ) , and ( 3 , ∞ ) .
Testing the Intervals Now we test each interval to see where the inequality 0"> x ( x − 3 ) > 0 holds. We pick a test value in each interval.
For the interval ( − ∞ , 0 ) , let's pick x = − 1 . Then 0"> x ( x − 3 ) = ( − 1 ) ( − 1 − 3 ) = ( − 1 ) ( − 4 ) = 4 > 0 . So the inequality holds in this interval.
For the interval ( 0 , 3 ) , let's pick x = 1 . Then x ( x − 3 ) = ( 1 ) ( 1 − 3 ) = ( 1 ) ( − 2 ) = − 2 < 0 . So the inequality does not hold in this interval.
For the interval ( 3 , ∞ ) , let's pick x = 4 . Then 0"> x ( x − 3 ) = ( 4 ) ( 4 − 3 ) = ( 4 ) ( 1 ) = 4 > 0 . So the inequality holds in this interval.
Determining the Solution Since the inequality 0"> x ( x − 3 ) > 0 holds for x < 0 and 3"> x > 3 , the solution to the inequality is x < 0 or 3"> x > 3 .
Examples
Understanding inequalities like 0"> x ( x − 3 ) > 0 is crucial in many real-world applications. For instance, imagine a company's profit is modeled by the equation P ( x ) = x 2 − 3 x , where x represents the number of units sold. The company wants to know for what values of x the profit is positive, i.e., 0"> P ( x ) > 0 . Solving the inequality 0"> x ( x − 3 ) > 0 helps the company determine the sales levels needed to ensure profitability. This concept extends to various fields, including physics, engineering, and economics, where understanding when a function or expression is positive or negative is essential for making informed decisions.
