write in interval notation
2x^2 < or = 9x+5
To write the given inequality in interval notation, we first need to solve the inequality and find the values of x that satisfy the inequality.
Let's start by solving the inequality:
2x^2 ≤ 9x + 5
First, we move all the terms to one side to form a quadratic equation:
2x^2 - 9x - 5 ≤ 0
Next, we factor the quadratic equation, if possible. However, this equation cannot be factored nicely, so we need to use an alternative method. We can solve it by using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 2, b = -9, and c = -5. Substituting these values into the quadratic formula, we get:
x = (-(-9) ± √((-9)^2 - 4(2)(-5))) / (2(2))
Simplifying further:
x = (9 ± √(81 + 40)) / 4
x = (9 ± √121) / 4
x = (9 ± 11) / 4
So we have two solutions:
x1 = (9 + 11) / 4 = 20/4 = 5
x2 = (9 - 11) / 4 = -2/4 = -1/2
Now, let's determine the intervals where the inequality is true.
To do this, we can use a graphical approach or a sign chart. Since we already have the solutions, we can use the sign chart method.
We choose test points in three regions:
1. A value less than -1/2, such as -1
2. A value between -1/2 and 5, such as 0
3. A value greater than or equal to 5, such as 6
Evaluating the inequality at those test points:
For x = -1: 2(-1)^2 ≤ 9(-1) + 5 --> 2 ≤ -9 + 5 --> 2 ≤ -4 (false)
For x = 0: 2(0)^2 ≤ 9(0) + 5 --> 0 ≤ 0 + 5 --> 0 ≤ 5 (true)
For x = 6: 2(6)^2 ≤ 9(6) + 5 --> 72 ≤ 54 + 5 --> 72 ≤ 59 (false)
Based on the evaluations, we can determine the intervals:
1. Solution: x is between -1/2 and 5, inclusive: [-1/2, 5]
2. The solution does not include any other interval, as the inequality is not satisfied in other regions.
Therefore, we can write the given inequality in interval notation as:
[-1/2, 5]