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

To solve the inequality 0.5x^2 + 8x + 24 < 0, we can use various methods. One way to approach this is by using the quadratic formula.

Step 1: Set the expression equal to zero: 0.5x^2 + 8x + 24 = 0

Step 2: Apply the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 0.5, b = 8, and c = 24.

x = (-8 ± √(8^2 - 40.524)) / (2*0.5) x = (-8 ± √(64 - 48)) / 1 x = (-8 ± √16) / 1 x = (-8 ± 4) / 1

So, x = -12 or x = -4.

Step 3: Plot the solutions on a number line to create intervals.

On the number line, mark the points -12 and -4.

Step 4: Test intervals to determine the sign of the expression.

Choose a test point within each interval and substitute into the original inequality to determine the sign. We will use -10, -6, and 0 as test points.

When x = -10: 0.5(-10)^2 + 8(-10) + 24 = 50 - 80 + 24 = -6 Since the result is negative, this interval satisfies the inequality.

When x = -6: 0.5(-6)^2 + 8(-6) + 24 = 18 - 48 + 24 = -6 Again, the result is negative, so this interval satisfies the inequality.

When x = 0: 0.5(0)^2 + 8(0) + 24 = 0 + 0 + 24 = 24 The result is positive, so this interval does not satisfy the inequality.

Step 5: Determine the solution.

From step 4, we know that the inequality is satisfied for x values in the intervals (-∞, -12) and (-12, -4). Therefore, the solution to the inequality 0.5x^2 + 8x + 24 < 0 is x ∈ (-∞, -12) ∪ (-12, -4).