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solve the Math: Polynomial inequality and express the solution in set notation.
2t^2 - 3 < or = to t
Inequalities involving parabolas can get interesting.
Solve as an equation:
2t^2 - t -3 = 0
(t+1)(2t-3) = 0
t = -1 or 3/2
Think now of the graphs of 2t^2 - 3 and t
You want the region where the parabola is below the line. That is,
-1 <= x <= 3/2
Of course, you can do analysis of the factors, considering positive and negative values, but it's a lot more work, and gets you the same result.
To solve the polynomial inequality, we will follow these steps:
Step 1: Move all terms to one side of the inequality to have a quadratic polynomial less than or equal to zero.
2t^2 - 3 - t ≤ 0
Step 2: Combine like terms on the left side of the inequality to obtain a quadratic polynomial in standard form.
2t^2 - t - 3 ≤ 0
Step 3: Solve the quadratic equation by factoring or using the quadratic formula. However, this equation cannot be easily factored, so we will use the quadratic formula.
The quadratic formula is given by: t = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 2, b = -1, and c = -3. Now we can substitute these values into the quadratic formula:
t = (-(-1) ± √((-1)^2 - 4(2)(-3))) / (2(2))
= (1 ± √(1 + 24)) / 4
= (1 ± √25) / 4
= (1 ± 5) / 4
This gives us two possible solutions:
t1 = (1 + 5) / 4 = 6 / 4 = 3/2
t2 = (1 - 5) / 4 = -4 / 4 = -1
Step 4: Determine the solution set by checking the intervals between the roots and the points outside those intervals. Since the inequality is less than or equal to, we include the solutions.
t ≤ -1 or t ≥ 3/2
Therefore, the solution in set notation is:
t ∈ (-∞, -1] U [3/2, +∞)