4-|2x-3|<=9
How do I deal with a negative absolute?
It's always true? How would I show my work on this
4-|2x-3| ≤ 9
-|2x-3| ≤ 5
|2x-3| ≥ -5
or by default:
|2x-3| ≥ 0 , since the absolute value of anything ≥ 0
2x-3 ≥ 0 OR -2x+3 ≥ 0
x ≥ 3/2
OR
-2x ≥ -3
x ≤ 3/2
so any value of x will work
the solution is the set of real numbers.
look at it this way..
in 4-|2x-3| ≤ 9
the left side has a maximum of 4, which is already ≤ 9
so no matter what we subtract, it can only get smaller, thus it will always be ≤ 9
To deal with a negative absolute value in an inequality, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.
Let's solve the given inequality step by step:
1. Start by splitting the inequality into two cases:
a. When the expression inside the absolute value is positive: 2x-3 >= 0
b. When the expression inside the absolute value is negative: 2x-3 < 0
2. Solve case (a) when 2x-3 >= 0:
Add 3 to both sides of the inequality: 2x >= 3
Divide both sides by 2 because we are solving for x: x >= 3/2
3. Solve case (b) when 2x-3 < 0:
Add 3 to both sides of the inequality: 2x < 3
Divide both sides by 2: x < 3/2
4. Combine the results from both cases:
The solution x for the given inequality is x <= 3/2 and x >= 3/2, which means any value between or including 3/2 is a valid solution.
Therefore, the solution to the inequality 4-|2x-3| <= 9 is x <= 3/2 and x >= 3/2.