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.
please help
abs(anything) is always greater than 0, which is greater than -4
1. An example of an absolute value inequality whose solution set is the empty set is |x + 1| < -2. Let me explain how we can determine this.
To solve absolute value inequalities, we need to consider two cases: when the expression inside the absolute value (in this case, x + 1) is positive and when it is negative. The absolute value of a number is always non-negative, so if the expression inside the absolute value is negative, the absolute value of it will be positive.
In the case of |x + 1| < -2, we need to find values of x for which the absolute value of (x + 1) is less than -2. However, since the absolute value is always non-negative, it can never be less than a negative number like -2. Therefore, there are no values of x that satisfy this condition, and the solution set is empty.
2. The inequality given is x + 2 ≥ -4. Let's explain why its solution set is all real numbers.
To find the solution set, we can start by isolating the variable on one side of the inequality. Subtracting 2 from both sides, we have x ≥ -4 - 2, which simplifies to x ≥ -6.
In this inequality, there are no restrictions or limitations on the values of x. Any real number that is greater than or equal to -6 will satisfy this inequality. Therefore, the solution set is all real numbers.
To understand this conceptually, we can think of the expression x + 2 as a line on the number line. If we sketch the line, we can see that any value of x that is greater than or equal to -6 will fall on or to the right of this line. Since the line extends indefinitely in both directions, the solution set encompasses all real numbers.