The following interval represents the solution of an absolute value inequality. Find an example of an absolute value inequality whose solution is described by this interval. [-2,1] please expalin
|x + 0.5| < 1.5
because the center of the interval is at -0.5 and the limits are 1.5 from the center.
To find an example of an absolute value inequality that has the solution interval [-2,1], we need to understand how absolute value inequalities work and how to represent them.
An absolute value inequality involves the absolute value of a variable or expression. In general, the form of an absolute value inequality is |x - a| ≤ b or |x - a| ≥ b, where "x" is the variable, "a" is a constant, and "b" is a positive number denoting the distance from the variable to the constant.
In our case, the solution interval is [-2,1]. This means that any value of "x" that satisfies the inequality will fall within the range from -2 to 1, inclusive. Let's break down the process of finding an absolute value inequality with this solution interval:
1. Determine the midpoint of the interval: The midpoint of the interval [-2,1] can be found by adding the two endpoints and dividing by 2: (-2 + 1) ÷ 2 = -0.5.
2. Determine the range of the interval: To find the range, subtract the smaller endpoint from the larger endpoint: 1 - (-2) = 3.
3. Set up the absolute value inequality: We want the absolute value inequality to have its solution within the range of -2 to 1. To achieve this, we can set the constant "a" in the inequality as the midpoint (-0.5), and set the positive number "b" as half the range (3 ÷ 2 = 1.5). Since we want the solution to be within the interval, we choose the less-than-or-equal-to sign: |x + 0.5| ≤ 1.5.
Therefore, an example of an absolute value inequality whose solution is described by the interval [-2,1] is |x + 0.5| ≤ 1.5. This inequality states that the absolute value of x plus 0.5 is less than or equal to 1.5, and the solution to this inequality will consist of all values of x that fall within the interval [-2,1].