solve the following polynomial and rational inequalities
2x^2+x-6>0
i need some help with this
2x^2+x-6>0
A=2
B=1
C=-6
Multiply A and C together
A=2, C=-6, 2x(-6)=-12
Find factors of -12 that add up to give you a positive value of 1, B
We figure out that factors of -12 that add up to give us 1 are 4 and -3
We rewrite our equation as:
2x^2+(4x-3x)-6>0
We can group our first two and our second two and factor those.
(2x^2+4x)-(3x-6)
We factor out a 2x from first grouping
2x(x+2)
We factor out a -3 from second grouping
-3(x+2)
What is left in both groupings must be the same to complete problem
Combine our GCF's and what left as our factors
(2x-3)(x+2)>0 Solve for x on both factors.
x>(3/2) x>-2
To solve the inequality 2x^2 + x - 6 > 0, you can use a combination of factoring and the sign chart method. Here's how you can approach it:
Step 1: Factor the quadratic expression if possible.
Start by trying to factor the quadratic expression 2x^2 + x - 6. Since the leading coefficient is 2, the factors of 2 need to be considered. However, in this case, the quadratic expression cannot be factored easily.
Step 2: Find the critical points.
To find the x-values where the inequality might change sign, set the quadratic expression equal to 0 and solve for x. This will give you the critical points.
Setting 2x^2 + x - 6 = 0, you can use factoring, completing the square, or the quadratic formula to find the solutions. In this case, using the quadratic formula is the easiest method.
The quadratic formula is x = (-b ± √(b^2 - 4ac))/2a.
For 2x^2 + x - 6 = 0, the coefficients are a = 2, b = 1, and c = -6. Plugging these values into the quadratic formula, you get:
x = (-1 ± √(1^2 - 4(2)(-6)))/(2(2))
x = (-1 ± √(1 + 48))/4
x = (-1 ± √49)/4
x = (-1 ± 7)/4
So, x = (6/4) or x = (-8/4).
The critical points are x = 3/2 and x = -2.
Step 3: Create a sign chart.
Make a sign chart by plotting the critical points on a number line.
-2 3/2
|------|--------------------|
Step 4: Test a value in each interval.
Choose a test value within each interval created by the critical points. Substitute the values into the original inequality to determine whether they satisfy the inequality or not.
For example, test x = -3, which is a value less than -2. Substitute it into the original inequality and see if it holds true:
2x^2 + x - 6 > 0
2(-3)^2 + (-3) - 6 > 0
18 - 3 - 6 > 0
9 > 0
Since 9 is indeed greater than 0, this means that the inequality holds true for this interval.
Now test a value between -2 and 3/2; let's choose x = 0:
2x^2 + x - 6 > 0
2(0)^2 + (0) - 6 > 0
0 - 6 > 0
-6 > 0
Since -6 is not greater than 0, this means the inequality does not hold true for this interval.
Next, test a value greater than 3/2; let's choose x = 2:
2x^2 + x - 6 > 0
2(2)^2 + (2) - 6 > 0
8 + 2 - 6 > 0
4 > 0
Since 4 is greater than 0, this means that the inequality holds true for this interval.
Step 5: Determine the solution.
Based on the sign chart, the solution to the inequality 2x^2 + x - 6 > 0 is x < -2 or x > 3/2.
Therefore, the solution set can be written as (-∞, -2) ∪ (3/2, ∞).