12x²+3x-2>0
Use Inspection of Sign Method (show full process)
(nonsense answer - report)
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12x²+3x-2>0
Use Inspection of Sign Method (show full process)
(nonsense answer - report)
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Answer:
To solve the inequality 12x² + 3x - 2 > 0 using the Inspection of Sign method, we need to determine the intervals where the expression is positive.
Step 1: Factorize the quadratic equation if possible.
The given quadratic equation cannot be easily factorized, so we'll proceed with the quadratic formula or completing the square.
Step 2: Find the critical points.
To find the critical points, we set the quadratic equation equal to zero and solve for x:
12x² + 3x - 2 = 0
Using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 12, b = 3, and c = -2.
x = (-3 ± √(3² - 4(12)(-2))) / (2(12))
x = (-3 ± √(9 + 96)) / 24
x = (-3 ± √105) / 24
So the critical points are x = (-3 + √105) / 24 and x = (-3 - √105) / 24.
Step 3: Plot the critical points on a number line.
We have two critical points: (-3 + √105) / 24 and (-3 - √105) / 24.
Step 4: Test the intervals.
We will test the intervals created by the critical points to determine the sign of the expression 12x² + 3x - 2.
For x < (-3 - √105) / 24:
Let's choose x = -1 as a test point.
Plugging x = -1 into the expression: 12(-1)² + 3(-1) - 2 = 12 - 3 - 2 = 7 > 0
Since the expression is positive for x = -1, this interval satisfies the inequality.
For (-3 - √105) / 24 < x < (-3 + √105) / 24:
Let's choose x = 0 as a test point.
Plugging x = 0 into the expression: 12(0)² + 3(0) - 2 = -2 < 0
Since the expression is negative for x = 0, this interval does not satisfy the inequality.
For x > (-3 + √105) / 24:
Let's choose x = 1 as a test point.
Plugging x = 1 into the expression: 12(1)² + 3(1) - 2 = 23 > 0
Since the expression is positive for x = 1, this interval satisfies the inequality.
Step 5: Write the solution.
Based on the inspection of sign method, the solution to the inequality 12x² + 3x - 2 > 0 is:
x < (-3 - √105) / 24 or x > (-3 + √105) / 24.