use a sign chart to solve inequality. Express answers in interval notation.
x^2+6<2x
x^2-2x+6 < 0
no solutions, which means that
x^2+x > 2x for all x
http://www.wolframalpha.com/input/?i=plot+y%3Dx^2%2B6%2Cy%3D2x+where+x%3D-3..3
Not sure how your "sign chart" method works, I will solve it the traditional way.
x^2 - 2x +6 < 0
Just looking at the graph of
y = x^2 - 2x + 6
we can tell that there is no solution.
http://www.wolframalpha.com/input/?i=Plot+y+%3D+x%5E2+-+2x+%2B+6
To solve the inequality x^2 + 6 < 2x using a sign chart, follow these steps:
1. Start by rewriting the inequality in standard form:
x^2 - 2x + 6 < 0
2. Create a sign chart or a number line. Divide the number line into three regions using the points that make the inequality equal to zero:
x^2 - 2x + 6 = 0
Solve this quadratic equation to find the x-intercepts.
The discriminant, b^2 - 4ac, is 4 - 4(1)(6) = -20, which is negative. This means the equation has no real solutions, so the quadratic function does not intersect the x-axis. Thus, there are no x-intercepts.
3. Choose test points within each region:
In the first region, choose a point to the left of the leftmost x-intercept as the test point (e.g., x = -10).
In the second region, choose a point between the two hypothetical x-intercepts as the test point (e.g., x = 0).
In the third region, choose a point to the right of the rightmost x-intercept as the test point (e.g., x = 10).
4. Substitute the chosen test points back into the original inequality and determine the sign of each region:
For x = -10: (-10)^2 - 2(-10) + 6 = 146, which is positive.
For x = 0: (0)^2 - 2(0) + 6 = 6, which is positive.
For x = 10: (10)^2 - 2(10) + 6 = 86, which is positive.
Therefore, the signs are positive for all three regions.
5. Determine the solution to the inequality:
Since the inequality is less than zero, we only consider the portions where the sign is negative. From the sign chart, we can see that there are no negative signs. This means there are no solutions to the inequality x^2 + 6 < 2x.
In interval notation, we can write the solution as an empty set:
∅ or {}
To solve the inequality x^2 + 6 < 2x, we can use a sign chart. Here's how you can do it:
1. Start by rewriting the inequality in standard form, with all terms on one side: x^2 - 2x + 6 < 0.
2. To create the sign chart, you need to find the critical points of the inequality, which are the values of x where the left side of the inequality changes its sign.
3. Set the left side of the inequality equal to zero: x^2 - 2x + 6 = 0. This quadratic equation does not factor, so we need to use the quadratic formula.
The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a.
For this equation, a = 1, b = -2, and c = 6. Substituting these values into the quadratic formula, we get:
x = (2 ± √((-2)^2 - 4(1)(6))) / 2(1)
x = (2 ± √(4 - 24)) / 2
x = (2 ± √(-20)) / 2
Since there is a negative value under the square root, the equation has no real solutions. Therefore, there are no critical points in this case.
4. Now, let's choose a test point within each interval determined by the critical points (although in this case, we only have one interval).
Let's test x = 0, which is a value between negative infinity and positive infinity. Substitute this value into the original inequality: (0)^2 + 6 < 2(0). Simplifying, we get 6 < 0, which is false.
Since the inequality is false for x = 0, it means that the inequality does not hold true for any value of x. Therefore, there are no solutions.
Graphically, this can also be visualized as the curve of the parabola x^2 - 2x + 6 being above the x-axis, indicating that there is no intersection with the line y = 0 (which represents the solution to the inequality).
Hence, the answer to the inequality x^2 + 6 < 2x, expressed in interval notation, is the empty set: {}.