Find the range of values of [tex]$x$[/tex] for which [tex]$2 x^2+7 x-15>0$[/tex]

Factor the quadratic 0"> 2 x 2 + 7 x − 15 > 0 to find roots x = − 5 and x = 1.5 , and determine that the inequality holds for x < − 5 or 1.5"> x > 1.5 .
Factor the quadratic 0"> x 2 − 2 x − 8 > 0 to find roots x = − 2 and x = 4 , and determine that the inequality holds for x < − 2 or 4"> x > 4 .
The solution to 0"> 2 x 2 + 7 x − 15 > 0 is x < − 5 or 1.5"> x > 1.5 .
The solution to 0"> x 2 − 2 x − 8 > 0 is x < − 2 or 4"> x > 4 .

Explanation

Problem Analysis We are given two quadratic inequalities to solve:

0"> 2 x 2 + 7 x − 15 > 0

0"> x 2 − 2 x − 8 > 0


We will solve each inequality separately by first finding the roots of the corresponding quadratic equation and then determining the intervals where the inequality holds.

Factoring the Quadratic For the first inequality, 0"> 2 x 2 + 7 x − 15 > 0 , we first find the roots of the equation 2 x 2 + 7 x − 15 = 0 .

We can factor the quadratic expression as follows:
2 x 2 + 7 x − 15 = ( 2 x − 3 ) ( x + 5 )

Finding the Roots Setting each factor to zero, we get the roots:

2 x − 3 = 0 ⟹ x = 2 3 ​ = 1.5 x + 5 = 0 ⟹ x = − 5
So, the roots are x = − 5 and x = 1.5 .

Determining the Intervals Now we analyze the sign of the quadratic expression in the intervals determined by the roots. Since the coefficient of x 2 is positive, the parabola opens upwards. Thus, the quadratic expression is positive for x < − 5 and 1.5"> x > 1.5 .

Therefore, the solution to the inequality 0"> 2 x 2 + 7 x − 15 > 0 is x < − 5 or 1.5"> x > 1.5 .

Factoring the Quadratic For the second inequality, 0"> x 2 − 2 x − 8 > 0 , we first find the roots of the equation x 2 − 2 x − 8 = 0 .

We can factor the quadratic expression as follows:
x 2 − 2 x − 8 = ( x − 4 ) ( x + 2 )

Finding the Roots Setting each factor to zero, we get the roots:

x − 4 = 0 ⟹ x = 4 x + 2 = 0 ⟹ x = − 2
So, the roots are x = − 2 and x = 4 .

Determining the Intervals Now we analyze the sign of the quadratic expression in the intervals determined by the roots. Since the coefficient of x 2 is positive, the parabola opens upwards. Thus, the quadratic expression is positive for x < − 2 and 4"> x > 4 .

Therefore, the solution to the inequality 0"> x 2 − 2 x − 8 > 0 is x < − 2 or 4"> x > 4 .

Final Answer In summary:

For 0"> 2 x 2 + 7 x − 15 > 0 , the solution is x < − 5 or 1.5"> x > 1.5 .
For 0"> x 2 − 2 x − 8 > 0 , the solution is x < − 2 or 4"> x > 4 .
Examples
Understanding quadratic inequalities is crucial in various real-world applications, such as optimization problems in business. For instance, a company might want to determine the range of production levels that yield a profit above a certain threshold. By modeling the profit as a quadratic function of production quantity, solving a quadratic inequality can identify the production ranges that meet the desired profit target. This helps in making informed decisions about production planning and resource allocation.

Answered by GinnyAnswer | 2025-07-03

The solution to the inequality 0"> 2 x 2 + 7 x − 15 > 0 is found by determining the roots and testing intervals. The final result indicates that the inequality holds for x < − 5 or 1.5"> x > 1.5 . Hence, the range of values is ( − ∞ , − 5 ) ∪ ( 1.5 , ∞ ) .
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Answered by Anonymous | 2025-07-04