Solve each inequality algebraically.
|2x-1|-2<3 I don't understand how the answer is -2<x<3
There are two important points involved:
1. When an inequality is multiplied or divided by a negative number on each side, the inequality changes direction.
For example:
4<5
-4>-5
2. When the absolute function comes into play, we have two conditions which have to be considered separately:
a. the contents of the absolute function is positive, and
b. the content of the absolute function is negative.
This means we double the number of conditions.
For example, to solve
|x-1|>4
we have
a. x-1>4, => x>5, and
b. -(x-1) > 4, therefore
(x-1) <-4
x<-3
Apply the same way to the given inequation and you should arrive at
-2<x<3 as the answer.
To solve the inequality |2x-1|-2<3 algebraically, we'll follow these steps:
Step 1: Remove the absolute value signs.
We start by removing the absolute value signs by considering two cases - when the expression inside the absolute value is positive and when it is negative.
Case 1: (2x-1) is positive.
In this case, we can drop the absolute value signs and rewrite the inequality as:
2x-1-2<3
Simplifying this inequality, we get:
2x-3<3
Step 2: Solve for x.
Next, we solve for x by isolating it on one side of the inequality. Adding 3 to both sides, we have:
2x<6
Dividing by 2, we get:
x<3
Case 2: (2x-1) is negative.
In this case, we must consider the opposite of the expression inside the absolute value. That means rewriting the inequality as:
-(2x-1)-2<3
Simplifying this inequality, we have:
-2x+1-2<3
Again, simplifying:
-2x-1<3
Step 3: Solve for x.
Isolate x on one side of the inequality. Adding 1 to both sides, we have:
-2x<4
Dividing by -2 is a little different. However, since dividing by -2 reverses the inequality sign, the direction stays the same.
So, we get:
x>-2
Step 4: Combine the solutions.
Now that we have found the solutions for both cases, we can combine them to find the final solution to the inequality.
The solution for Case 1 is: x<3
The solution for Case 2 is: x>-2
Therefore, the combined solution is: -2 < x < 3