Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation.
[tex]x^2-9 x+14\ \textgreater \ 0[/tex]
Answer (1)
Factor the quadratic expression: x 2 − 9 x + 14 = ( x − 2 ) ( x − 7 ) .
Find the roots: x = 2 and x = 7 .
Test intervals ( − ∞ , 2 ) , ( 2 , 7 ) , and ( 7 , ∞ ) to determine where 0"> ( x − 2 ) ( x − 7 ) > 0 .
Express the solution in interval notation: ( − ∞ , 2 ) ∪ ( 7 , ∞ ) .
Explanation
Problem Analysis We are given the polynomial inequality 0"> x 2 − 9 x + 14 > 0 . Our goal is to find the values of x that satisfy this inequality, express the solution in interval notation, and represent it graphically on a number line.
Factoring the Quadratic To solve the inequality, we first factor the quadratic expression: x 2 − 9 x + 14 = ( x − 2 ) ( x − 7 ) So, the inequality becomes: 0"> ( x − 2 ) ( x − 7 ) > 0
Finding Critical Points The roots of the quadratic equation ( x − 2 ) ( x − 7 ) = 0 are x = 2 and x = 7 . These roots divide the real number line into three intervals: ( − ∞ , 2 ) , ( 2 , 7 ) , and ( 7 , ∞ ) . We need to determine the sign of the expression ( x − 2 ) ( x − 7 ) in each interval.
Testing Intervals Let's test a value in each interval:
Interval ( − ∞ , 2 ) : Choose x = 0 . Then 0"> ( 0 − 2 ) ( 0 − 7 ) = ( − 2 ) ( − 7 ) = 14 > 0 . So, the expression is positive in this interval.
Interval ( 2 , 7 ) : Choose x = 3 . Then ( 3 − 2 ) ( 3 − 7 ) = ( 1 ) ( − 4 ) = − 4 < 0 . So, the expression is negative in this interval.
Interval ( 7 , ∞ ) : Choose x = 8 . Then 0"> ( 8 − 2 ) ( 8 − 7 ) = ( 6 ) ( 1 ) = 6 > 0 . So, the expression is positive in this interval.
Solution Set Since we want to find where 0"> ( x − 2 ) ( x − 7 ) > 0 , we are looking for the intervals where the expression is positive. From our tests, these are ( − ∞ , 2 ) and ( 7 , ∞ ) . Therefore, the solution set is: ( − ∞ , 2 ) ∪ ( 7 , ∞ )
Final Answer The solution set in interval notation is ( − ∞ , 2 ) ∪ ( 7 , ∞ ) . This means that x can be any value less than 2 or greater than 7. We can represent this graphically on a number line by shading the regions to the left of 2 and to the right of 7, with open circles at 2 and 7 to indicate that these values are not included in the solution.
Examples
Understanding polynomial inequalities is crucial in various fields, such as physics and engineering. For instance, when designing a bridge, engineers need to ensure that the structure can withstand certain loads. This often involves solving inequalities to determine the range of acceptable stress levels. Similarly, in physics, analyzing the motion of objects under certain conditions may require solving polynomial inequalities to find the time intervals during which specific conditions are met. These applications highlight the practical importance of mastering polynomial inequalities.
