Question

Which one of the following intervals is not symmetric about \(x=5 ?\) a. (1,9) b. (4,6)\(\quad\) c. (3,8) d. (4.5,5.5)

Short answer

a) (1,9) b) (4,6) c) (3,8) d) (4.5,5.5) Answer: c) (3,8)

Step-by-step solution

Check the interval (1,9)

Calculate the distance between 5 and the interval endpoints 1 and 9, respectively: \((5-1) = 4\) and \((9-5) = 4\). Since both distances are equal, this interval is symmetric about \(x=5\).

Check the interval (4,6)

Calculate the distance between 5 and the interval endpoints 4 and 6, respectively: \((5-4) = 1\) and \((6-5) = 1\). Since both distances are equal, this interval is symmetric about \(x=5\).

Check the interval (3,8)

Calculate the distance between 5 and the interval endpoints 3 and 8, respectively: \((5-3) = 2\) and \((8-5) = 3\). The distances are not equal, so this interval is not symmetric about \(x=5\).

Check the interval (4.5,5.5)

Calculate the distance between 5 and the interval endpoints 4.5 and 5.5, respectively: \((5-4.5) = 0.5\) and \((5.5-5) = 0.5\). Since both distances are equal, this interval is symmetric about \(x=5\).

Conclusion

The interval (3,8) is not symmetric about \(x=5\), so the correct answer is option c.

Key concepts

Distance Calculation

Understanding distance calculation is key to solving problems related to symmetric intervals. When we talk about distance in the realm of mathematics, we refer to how far apart two points are on a number line. For a specific interval to be symmetric around a given point, like in our exercise where intervals are checked for symmetry around \(x = 5\), it's critical to calculate the distance of this midpoint from both ends of the interval. For example, if you have an interval (1,9), you measure how far \(x=5\) is from 1 and 9 separately. You do this by calculating \((5-1) = 4\) and \((9-5) = 4\). Both distances being equal indicates symmetry around \(x=5\). Consistently applying these calculations gives a clear indication of symmetry in any interval.Keep in mind the importance of accuracy in these calculations. Incorrect distance measurements can lead you to wrong conclusions about interval symmetry.

Interval Symmetry

Interval symmetry occurs when a number, in this case, \(x=5\), divides an interval into two equal parts. This concept is crucial when determining symmetric intervals. For an interval to be considered symmetric about a number, say \(x=5\), the distances from \(x=5\) to both endpoints should equal.Let’s explore the interval \( (3,8) \). Here, the distances from \(x = 5\) to both endpoints are different: from 3 is \((5-3) = 2\) and from 8 is \((8-5) = 3\). The unequal distances mean the interval is not symmetric around \(x=5\). To simplify:

• Identify your midpoint, here \(x=5\).
• Subtract the midpoint from each endpoint of the interval.
• Compare the distances. Equal distances mean symmetry.

This method can effortlessly be applied to any interval.

Coordinate Geometry

In coordinate geometry, we often deal with the relationships of points and lines on a graph. Understanding these relationships can help in visualizing symmetry and distances intuitively. When considering this exercise, envisioning the intervals on a number line or graph can be immensely beneficial.Each endpoint of an interval corresponds to coordinates on a line. Symmetric intervals about a point \(x=5\) mean this point acts as a midpoint on that line. For instance, in the case of interval (4,6), these coordinates \(x=4\) and \(x=6\) will be equidistant from \(x=5\). Hence, on a graph, these points will form a straight line stopping just as far in one direction as the other.It's not just theoretical but a visual and spatial concept, using diagrams or sketches can help solidify understanding:

• Draw the number line and mark the midpoint (e.g., \(x=5\)).
• Mark the endpoints of the interval.
• Check if the midpoint is exactly between the endpoints.

Approaching the problem this way can make symmetric intervals and other geometric concepts feel intuitive.