0,5x^2 +8x + 24 <0

To solve the inequality 0.5x^2 + 8x + 24 < 0, we need to find the values of x that satisfy the inequality. We can do this by considering the quadratic equation 0.5x^2 + 8x + 24 = 0.

To start, let's find the roots of the equation by using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a) For our equation, a = 0.5, b = 8, and c = 24.

x = (-8 ± √(8^2 - 4(0.5)(24))) / (2(0.5)) x = (-8 ± √(64 - 48)) / 1 x = (-8 ± √16) / 1 x = (-8 ± 4) / 1

So we have two possible values for x: x1 = (-8 + 4) = -4 x2 = (-8 - 4) = -12

We can now plot the quadratic equation on a number line with these two values:

----|-----|-----|----- -12 -4

Now we need to test the intervals between these values to find the solution to the inequality.

For x < -12, let's choose x = -13 as a test value: 0.5*(-13)^2 + 8*(-13) + 24 = 169/2 - 104 + 24 = 169/2 - 104/2 + 48/2 = 113/2 > 0

Since the value is greater than 0, this interval is not a solution.

For -12 < x < -4, let's choose x = -5 as a test value: 0.5*(-5)^2 + 8*(-5) + 24 = 25/2 - 40 + 24 = 25/2 - 40/2 + 48/2 = 33/2 > 0

Again, the value is greater than 0, so this interval is also not a solution.

For x > -4, let's choose x = 0 as a test value: 0.5*(0)^2 + 8*(0) + 24 = 24 > 0

The value is greater than 0, so this interval is not a solution either.

From this analysis, we can conclude that there are no values of x that satisfy the inequality 0.5x^2 + 8x + 24 < 0.