Answer: I can help you with this problem. To write and solve an absolute value inequality, you need to use the definition of absolute value and the properties of inequalities. Here are the steps:
The absolute value of a number is its distance from zero on the number line. For example, | 3 | = 3 and | -3 | = 3, because both 3 and -3 are 3 units away from zero.
The distance between two numbers on the number line is the absolute value of their difference. For example, the distance between 6 and 10 is | 6 - 10 | = | -4 | = 4, and the distance between 6 and 2 is | 6 - 2 | = | 4 | = 4.
To write an absolute value inequality that describes all numbers whose distance from 6 is at most 12, we can use the variable x to represent any number, and write | x - 6 | ≤ 12. This means that the absolute value of x minus 6 is less than or equal to 12, which means that the distance between x and 6 is less than or equal to 12.
To solve an absolute value inequality, we need to split it into two cases: when the expression inside the absolute value is positive or negative. We can use a number line to help us visualize this. For example, if we have | x - 6 | ≤ 12, we can draw a number line with 6 in the middle, and mark the points that are 12 units away from 6 on both sides. These points are -6 and 18, as shown below.
The inequality | x - 6 | ≤ 12 means that x can be any number between -6 and 18, including -6 and 18. We can write this as -6 ≤ x ≤ 18. This is the solution in interval notation: [-6, 18].
Alternatively, we can solve an absolute value inequality algebraically, by using the following property: if | u | ≤ a, where u is any algebraic expression and a is any positive number, then -a ≤ u ≤ a. For example, if we have | x - 6 | ≤ 12, we can apply this property and get -12 ≤ x - 6 ≤ 12. Then, we can add 6 to all sides of the inequality and get -6 ≤ x ≤ 18. This is the same solution as before.