1. give an example of an absolute value inequality whose solution set is the empty set
2. explain why \x+2\ is greater than or = to -4 has all real numbers as its solution set.
can you please help me?
1. To find an example of an absolute value inequality whose solution set is the empty set, we need to understand how absolute value works. The absolute value of a number is its distance from zero on the number line, so it is always a non-negative value.
Let's consider the absolute value inequality |x + 5| < 0. In this case, the absolute value on the left side will always be greater than or equal to zero because it represents the distance of x + 5 from zero. However, it can never be less than zero. Therefore, there are no values of x that satisfy this inequality, leading to an empty solution set.
2. To understand why the inequality |x + 2| ≥ -4 has all real numbers as its solution set, we need to consider the definition of absolute value.
The absolute value of any real number is always non-negative, meaning it is equal to or greater than zero. In the given inequality, we have |x + 2| ≥ -4. Since the right side is -4, which is already a non-negative value, any value of x will make the left side of the inequality equal to or greater than -4.
Therefore, the inequality is true for all real numbers, as there is no restriction on the values of x. This is why the solution set of this inequality is all real numbers.