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

To solve the inequality 0.5x^2 + 8x + 24 < 0, we can use the quadratic formula or factor the quadratic expression. Let's factor it.

First, multiply the entire inequality by 2 to get rid of the decimal point:

1x^2 + 16x + 48 < 0

Now, let's factor the quadratic expression:

(x + 4)(x + 12) < 0

To find the values of x that will make the expression less than zero, we need to examine the sign changes. The expression changes sign when it crosses the x-axis.

Setting each factor to zero, we get:

x + 4 = 0 x = -4

x + 12 = 0 x = -12

Now, we can create an interval chart to determine when the expression is negative:

Interval: (-∞, -12) (-12, -4) (-4, ∞) Sign: negative positive negative

From the chart, we can see that the expression is negative in the interval (-12, -4).

Therefore, the solution to the inequality 0.5x^2 + 8x + 24 < 0 is:

-12 < x < -4