If 5x + x2 > 100, then x is not
10
08
07
09
The correct answer and explanation is :
The given inequality is:
[
5x + x^2 > 100
]
We need to determine for which values of ( x ), this inequality does not hold. In other words, we need to find the values of ( x ) that do not satisfy this inequality.
Step 1: Rewrite the inequality
Rearrange the given inequality:
[
x^2 + 5x – 100 > 0
]
This is a quadratic inequality of the form ( ax^2 + bx + c > 0 ), where ( a = 1 ), ( b = 5 ), and ( c = -100 ).
Step 2: Solve the corresponding quadratic equation
We now solve the related quadratic equation:
[
x^2 + 5x – 100 = 0
]
To solve for ( x ), we can use the quadratic formula:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
Substitute ( a = 1 ), ( b = 5 ), and ( c = -100 ) into the quadratic formula:
[
x = \frac{-5 \pm \sqrt{5^2 – 4(1)(-100)}}{2(1)}
]
[
x = \frac{-5 \pm \sqrt{25 + 400}}{2}
]
[
x = \frac{-5 \pm \sqrt{425}}{2}
]
The square root of 425 is approximately 20.615, so:
[
x = \frac{-5 \pm 20.615}{2}
]
This gives two solutions:
[
x_1 = \frac{-5 + 20.615}{2} \approx 7.81
]
[
x_2 = \frac{-5 – 20.615}{2} \approx -12.81
]
Thus, the roots of the equation are approximately ( x_1 \approx 7.81 ) and ( x_2 \approx -12.81 ).
Step 3: Analyze the inequality
Now that we have the roots, we can analyze the inequality ( x^2 + 5x – 100 > 0 ). The quadratic expression will be greater than 0 outside the interval defined by the roots. Specifically, the inequality will hold for:
[
x < -12.81 \quad \text{or} \quad x > 7.81
]
Therefore, the inequality is not satisfied for values of ( x ) in the interval ( -12.81 < x < 7.81 ).
Step 4: Check the given options
The options provided are:
- 10
- 08
- 07
- 09
Now, we check these values against the interval where the inequality does not hold (i.e., between ( -12.81 ) and ( 7.81 )):
- ( x = 10 ): This value is greater than 7.81, so it satisfies the inequality.
- ( x = 08 ): This value is greater than 7.81, so it satisfies the inequality.
- ( x = 07 ): This value is less than 7.81, so it does not satisfy the inequality.
- ( x = 09 ): This value is greater than 7.81, so it satisfies the inequality.
Conclusion
The value of ( x ) that does not satisfy the inequality ( 5x + x^2 > 100 ) is ( 07 ).
Therefore, the correct answer is ( \boxed{07} ).