Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation.
x³ ≥2x²
Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
B. The solution set is the empty set.
Answer (2)
The solution set of the inequality x 3 ≥ 2 x 2 is expressed in interval notation as [ 2 , ∞ ) ∪ { 0 } . This includes the points where the expression is zero as well as where it is positive. Therefore, the correct choice is option A.
;
Rewrite the inequality: x 3 − 2 x 2 ≥ 0 .
Factor the inequality: x 2 ( x − 2 ) ≥ 0 .
Find critical points: x = 0 and x = 2 .
Express the solution set in interval notation: [ 2 , ∞ ) ∪ { 0 } .
Explanation
Understanding the Problem We are given the polynomial inequality x 3 g e 2 x 2 . Our goal is to solve this inequality and express the solution set in interval notation.
Rewriting the Inequality First, we rewrite the inequality as x 3 − 2 x 2 g e 0 .
Factoring the Inequality Next, we factor the left side of the inequality: x 2 ( x − 2 ) g e 0 .
Finding Critical Points Now, we find the critical points by setting each factor to zero: x 2 = 0 and x − 2 = 0 . This gives x = 0 and x = 2 .
Testing Intervals We create a number line and test intervals determined by the critical points: ( − ∞ , 0 ) , ( 0 , 2 ) , and ( 2 , ∞ ) .
For the interval ( − ∞ , 0 ) , we test x = − 1 : ( − 1 ) 2 ( − 1 − 2 ) = 1 ( − 3 ) = − 3 < 0 .
For the interval ( 0 , 2 ) , we test x = 1 : ( 1 ) 2 ( 1 − 2 ) = 1 ( − 1 ) = − 1 < 0 .
For the interval ( 2 , ∞ ) , we test x = 3 : 0"> ( 3 ) 2 ( 3 − 2 ) = 9 ( 1 ) = 9 > 0 .
Determining the Solution Set Since we want x 2 ( x − 2 ) g e 0 , the solution set includes the intervals where the expression is positive or zero. The expression is zero when x = 0 or x = 2 , and positive when 2"> x > 2 . Therefore, the solution set is x = 0 and [ 2 , ∞ ) .
Expressing in Interval Notation Expressing the solution set in interval notation, we have [ 2 , ∞ ) ∪ { 0 } .
Final Answer Therefore, the solution set is [ 2 , ∞ ) ∪ { 0 } .
Examples
Polynomial inequalities are useful in various real-world applications, such as determining the range of values for a manufacturing process to ensure that the product meets certain quality standards. For example, if the profit of selling x items is given by P ( x ) = x 3 − 2 x 2 , we can use the inequality x 3 ≥ 2 x 2 to find the minimum number of items that need to be sold to ensure a non-negative profit. Solving this inequality helps businesses make informed decisions about production and sales targets.
