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Magazine Chapter 0: Problem 34
Solve each polynomial inequality and express the solution set in interval notation. $$3 t^{2} \geq-5 t+2$$
Short Answer
Expert verified
The solution set is \( (-\infty, -2] \cup [\frac{1}{3}, \infty) \).
Step by step solution
01
Rearrange the Inequality
First, we need to rewrite the inequality into a standard form polynomial inequality by bringing all terms to one side of the inequality sign. Start with the given inequality: \(3t^2 \geq -5t + 2\). Move all terms to the left side: \(3t^2 + 5t - 2 \geq 0\).
02
Solve the Corresponding Equation
To solve the inequality, first solve the corresponding equation by setting the expression equal to zero: \(3t^2 + 5t - 2 = 0\). Use the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = 5\), and \(c = -2\). Calculate the discriminant: \(b^2 - 4ac = 5^2 - 4(3)(-2) = 25 + 24 = 49\). Now find the roots: \(t = \frac{-5 \pm \sqrt{49}}{6}\). The roots are \(t = \frac{-5 + 7}{6} = \frac{1}{3}\) and \(t = \frac{-5 - 7}{6} = -2\).
03
Determine Intervals
With the roots \(t = \frac{1}{3}\) and \(t = -2\), divide the number line into intervals: \((-\infty, -2)\), \([-2, \frac{1}{3}]\), and \((\frac{1}{3}, \infty)\).
04
Test Each Interval
Choose a test point in each interval and substitute it into the inequality \(3t^2 + 5t - 2 \geq 0\): - For the interval \((-\infty, -2)\), choose \(t = -3\): \(3(-3)^2 + 5(-3) - 2 = 27 - 15 - 2 = 10\), which is greater than 0. - For the interval \([-2, \frac{1}{3}]\), choose \(t = 0\): \(3(0)^2 + 5(0) - 2 = -2\), which is less than 0. - For the interval \((\frac{1}{3}, \infty)\), choose \(t = 1\): \(3(1)^2 + 5(1) - 2 = 3 + 5 - 2 = 6\), which is greater than 0.
05
Write the Solution in Interval Notation
From the tests, the inequality holds for intervals \((-\infty, -2)\) and \((\frac{1}{3}, \infty)\). Since \(-2\) and \(\frac{1}{3}\) make the expression equal to zero and the inequality uses \(\geq\), include these points. So, the solution in interval notation is \((-\infty, -2] \cup [\frac{1}{3}, \infty)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a crucial tool for solving quadratic equations that are written in the form \(ax^2 + bx + c = 0\). This formula provides a systematic method to find the roots, also known as solutions or zeroes, of the equation. Here's how it works in simple steps:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Let's break it down further:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Let's break it down further:
- \(a\), \(b\), and \(c\) are coefficients from the equation \(ax^2 + bx + c = 0\).
- The symbol \(\pm\) means you'll get two roots: one by adding and one by subtracting the square root.
- The expression under the square root, \(b^2 - 4ac\), is known as the discriminant.
Interval Notation
Once we find the roots of a polynomial inequality, interval notation helps us express the parts of the number line where the inequality holds true. This notation provides a clear and concise way to communicate the ranges of values, which are solutions to the inequality.
Here鈥檚 how interval notation works:
Here鈥檚 how interval notation works:
- It consists of endpoints and parentheses or brackets.
- Use a parenthesis \(()\) when the endpoint is not included, which indicates an open interval.
- Use a bracket \([]\) if the endpoint is included, representing a closed interval.
- Combine different intervals using the union symbol \(\cup\) when a solution includes more than one interval.
Discriminant
The discriminant is a special expression within the quadratic formula. It significantly impacts the nature of the roots of a quadratic equation. The discriminant formula is given by \(b^2 - 4ac\). Depending on its value, you can predict the type and number of solutions an equation has.
Here鈥檚 what the discriminant tells us:
Here鈥檚 what the discriminant tells us:
- If \(b^2 - 4ac > 0\), there are two distinct real roots.
- If \(b^2 - 4ac = 0\), there is exactly one real root (also called a repeated or double root).
- If \(b^2 - 4ac < 0\), there are no real roots, only complex or imaginary ones.
