Solve the inequality algebraically
11x-3¡Ý-4x^2
Is ¡Ý supposed to be an inequality sign? Which one?
Solve:
x^3-9x^2+20x-20<0 write answer in the interval form.
To solve the inequality algebraically, we need to follow these steps:
Step 1: Move all the terms to one side of the inequality to create a quadratic equation. This can be done by adding 4x^2 to both sides and subtracting 11x from both sides:
4x^2 + 11x - 3 ≥ 0
Step 2: Rearrange the equation in standard quadratic form, where the highest order term comes first:
4x^2 + 11x - 3 ≥ 0 becomes 4x^2 + 11x - 3 = 0
Step 3: Factor the quadratic equation. However, in this specific case, the equation can't be factored easily, so we will use the quadratic formula instead.
The quadratic formula is: x = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 4, b = 11, and c = -3. Plugging these values into the quadratic formula, we get:
x = (-11 ± √(11^2 - 4 * 4 * -3))/(2 * 4)
Simplifying further:
x = (-11 ± √(121 + 48))/8
x = (-11 ± √169)/8
x = (-11 ± 13)/8
This gives us two possible solutions:
x = (-11 + 13)/8 = 2/8 = 1/4
x = (-11 - 13)/8 = -24/8 = -3
Step 4: Determine the intervals where the quadratic equation is greater than or equal to zero. We can use these solutions to create intervals on the number line.
From x = -∞ to x = -3, the equation is non-negative (greater than or equal to zero) since -3 is one of the solutions. We can represent this interval as (-∞, -3].
From x = 1/4 onwards, the equation is non-negative since 1/4 is also one of the solutions. We can represent this interval as [1/4, ∞).
Therefore, the solution to the inequality 11x - 3 ≥ -4x^2 is:
x ∈ (-∞, -3] U [1/4, ∞)