If 5x + x² > 100, then x is not
The correct answer and explanation is :
We are given the inequality: 5x+x2>1005x + x^2 > 100
We are to determine the values that x is not, meaning values of x that do not satisfy the inequality.
Step 1: Rewrite the inequality in standard form
x2+5x−100>0x^2 + 5x – 100 > 0
We want to solve this inequality. First, find the roots of the related quadratic equation: x2+5x−100=0x^2 + 5x – 100 = 0
Step 2: Solve the quadratic equation
Use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
For a=1a = 1, b=5b = 5, and c=−100c = -100: x=−5±52−4(1)(−100)2(1)=−5±25+4002=−5±4252x = \frac{-5 \pm \sqrt{5^2 – 4(1)(-100)}}{2(1)} = \frac{-5 \pm \sqrt{25 + 400}}{2} = \frac{-5 \pm \sqrt{425}}{2} 425≈20.6155⇒x≈−5±20.61552\sqrt{425} \approx 20.6155 \Rightarrow x \approx \frac{-5 \pm 20.6155}{2}
So the roots are: x1≈−5+20.61552≈7.81,x2≈−5−20.61552≈−12.81x_1 \approx \frac{-5 + 20.6155}{2} \approx 7.81, \quad x_2 \approx \frac{-5 – 20.6155}{2} \approx -12.81
Step 3: Analyze the inequality
We now analyze the inequality: x2+5x−100>0x^2 + 5x – 100 > 0
This expression is a parabola opening upward (since the coefficient of x2x^2 is positive). It will be greater than 0 outside the roots:
- When x<−12.81x < -12.81 or x>7.81x > 7.81, the expression is positive.
- When −12.81<x<7.81-12.81 < x < 7.81, the expression is negative.
Thus, the solution set is: x<−12.81orx>7.81x < -12.81 \quad \text{or} \quad x > 7.81
Final Answer:
x is not between −12.81 and 7.81\boxed{x \text{ is not between } -12.81 \text{ and } 7.81}
Or if presented in multiple choice:
x is not in the interval (−12.81,7.81)(-12.81, 7.81)
This means any x-value within that interval does not satisfy the inequality.