Solve for x. |4x - 1| >= 6; x \le or x \ge (Type integers or simplified fractions.)
There are two cases to consider:
Case 1: 4x - 1 is greater than or equal to 0 (|4x - 1| = 4x - 1)
In this case, the inequality becomes: 4x - 1 >= 6
Adding 1 to both sides, we have: 4x >= 7
Dividing both sides by 4, we get: x >= 7/4
Case 2: 4x - 1 is less than 0 (|4x - 1| = -(4x - 1) = -4x + 1)
In this case, the inequality becomes: -4x + 1 >= 6
Subtracting 1 from both sides, we have: -4x >= 5
Dividing both sides by -4 (and reversing the inequality), we get: x <= -5/4
Combining the two cases, the solution to the inequality is:
x <= -5/4 or x >= 7/4
To solve the inequality |4x - 1| >= 6, we can break it down into two separate inequalities:
1. 4x - 1 >= 6
2. -(4x - 1) >= 6
Let's solve them one by one:
1. Solving the first inequality:
4x - 1 >= 6
Add 1 to both sides:
4x >= 7
Divide both sides by 4 (as the coefficient of x):
x >= 7/4
2. Solving the second inequality:
-(4x - 1) >= 6
Simplify inside the absolute value:
-4x + 1 >= 6
Subtract 1 from both sides:
-4x >= 5
Divide both sides by -4 (note that dividing by a negative number reverses the inequality):
x <= -5/4
Combining both solutions, we have:
x <= -5/4 or x >= 7/4
To solve the inequality |4x - 1| >= 6, we need to consider two separate cases: when 4x - 1 is positive and when it is negative.
Case 1: 4x - 1 is positive:
In this case, the absolute value becomes unnecessary, and we can rewrite the inequality as:
4x - 1 >= 6
Now, let's solve for x:
4x - 1 >= 6
Add 1 to both sides:
4x >= 7
Divide both sides by 4:
x >= 7/4
So, for the positive case, x is greater than or equal to 7/4.
Case 2: 4x - 1 is negative:
In this case, the absolute value makes the expression positive, so we need to change the inequality to reflect that:
-(4x - 1) >= 6
Now, let's solve for x:
-(4x - 1) >= 6
Distribute the negative sign:
-4x + 1 >= 6
Subtract 1 from both sides:
-4x >= 5
Divide both sides by -4 (and remember that dividing by a negative number flips the inequality):
x <= 5/(-4)
x <= -5/4
So, for the negative case, x is less than or equal to -5/4.
Combining the two cases, the solution to the inequality |4x - 1| >= 6 is:
x <= -5/4 or x >= 7/4