Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation.
[tex]$3 x^2-2 x\ \textless \ 5$[/tex]
Answer (1)
Rewrite the inequality: 3 x 2 − 2 x − 5 < 0 .
Factor the quadratic: ( 3 x − 5 ) ( x + 1 ) = 0 , roots are x = − 1 and x = 3 5 .
Test intervals: ( − ∞ , − 1 ) , ( − 1 , 3 5 ) , and ( 3 5 , ∞ ) .
Express the solution in interval notation: ( − 1 , 3 5 ) .
Explanation
Rewrite the inequality First, we need to rewrite the inequality in the standard form a x 2 + b x + c < 0 . Subtract 5 from both sides of the inequality 3 x 2 − 2 x < 5 to get 3 x 2 − 2 x − 5 < 0 .
Find the roots Next, we find the roots of the corresponding quadratic equation 3 x 2 − 2 x − 5 = 0 . We can factor this quadratic as ( 3 x − 5 ) ( x + 1 ) = 0 . Thus, the roots are x = − 1 and x = 3 5 .
Determine the intervals Now, we determine the intervals determined by the roots. The roots are x = − 1 and x = 3 5 . These roots divide the real number line into three intervals: ( − ∞ , − 1 ) , ( − 1 , 3 5 ) , and ( 3 5 , ∞ ) .
Test each interval We test a value from each interval in the original inequality 3 x 2 − 2 x − 5 < 0 to determine where the inequality is true.
For the interval ( − ∞ , − 1 ) , let's test x = − 2 : 0"> 3 ( − 2 ) 2 − 2 ( − 2 ) − 5 = 12 + 4 − 5 = 11 > 0 . So the inequality is not true in this interval.
For the interval ( − 1 , 3 5 ) , let's test x = 0 : 3 ( 0 ) 2 − 2 ( 0 ) − 5 = − 5 < 0 . So the inequality is true in this interval.
For the interval ( 3 5 , ∞ ) , let's test x = 2 : 0"> 3 ( 2 ) 2 − 2 ( 2 ) − 5 = 12 − 4 − 5 = 3 > 0 . So the inequality is not true in this interval.
Express the solution in interval notation The solution set is the interval where the inequality is true, which is ( − 1 , 3 5 ) .
Final Answer The solution set in interval notation is ( − 1 , 3 5 ) .
Examples
Polynomial inequalities are useful in various real-world applications. For example, in business, a company might model its profit as a quadratic function of the number of units sold. By solving a polynomial inequality, the company can determine the range of sales that will result in a positive profit. Similarly, in physics, projectile motion can be modeled using quadratic equations, and solving inequalities can help determine the range of launch angles that will result in the projectile reaching a certain height or distance. Understanding polynomial inequalities allows for optimizing outcomes and making informed decisions in these scenarios.
