Solve the inequality. Round your answer to the nearest tenth.
x2 + 5⋅x - 7 > 5
To solve the inequality x^2 + 5x - 7 > 5, we can follow these steps:
1. Start by moving all the terms to one side of the inequality to have a quadratic expression:
x^2 + 5x - 7 - 5 > 0
Simplifying, we get:
x^2 + 5x - 12 > 0
2. Factorize the quadratic equation if possible. In this case, the equation cannot be factored easily, so we can proceed to the next step.
3. Use the quadratic formula to find the roots of the equation. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the roots are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = 5, and c = -12. Plugging in these values, we get:
x = (-5 ± √(5^2 - 4(1)(-12))) / (2(1))
Simplifying further:
x = (-5 ± √(25 + 48)) / 2
x = (-5 ± √73) / 2
4. Now we need to determine the intervals where the quadratic expression is greater than zero. To do this, we consider the values of x that make the expression positive (greater than zero).
5. Let's find the critical points by setting the expression equal to zero:
x = (-5 ± √73) / 2
6. Plot these values on a number line:
Critical points: -5 - √73/2 and -5 + √73/2
-5 - √73/2 -5 + √73/2
7. Now we test the intervals between the critical points to determine the sign of the expression and whether it is greater than zero or not. We can choose any value within each interval to test. Let's pick -6 and 0:
For x < -5 - √73/2: Let's substitute x = -6 into the expression:
(-6)^2 + 5(-6) - 12 = 0
Since this value is equal to zero, the expression is not greater than zero in this interval.
For -5 - √73/2 < x < -5 + √73/2: Let's substitute x = 0 into the expression:
(0)^2 + 5(0) - 12 = -12
Since this value is negative, the expression is not greater than zero in this interval.
For x > -5 + √73/2: Let's substitute x = 6 into the expression:
(6)^2 + 5(6) - 12 = 54
Since this value is greater than zero, the expression is greater than zero in this interval.
8. Therefore, the solution to the inequality x^2 + 5x - 7 > 5 is:
x > -5 + √73/2
Rounding to the nearest tenth: x > -0.8