Solve |1 - 4x| > 7
{x|x < -1.5 ∪ x > 1.5}
{x|x < -1.5 ∪ x > 2}
{x|-1.5 < x < 2}
The given inequality is |1 - 4x| > 7.
We can solve this inequality by breaking it down into two inequalities:
1 - 4x > 7 (when 1 - 4x is positive)
-(1 - 4x) > 7 (when 1 - 4x is negative)
Solving the first inequality:
1 - 4x > 7
-4x > 6
x < -1.5
Solving the second inequality:
-(1 - 4x) > 7
1 - 4x < -7
-4x < -8
x > 2
Therefore, the solution to the inequality |1 - 4x| > 7 is:
x < -1.5 or x > 2
To solve the inequality |1 - 4x| > 7, we can break it down into two separate inequalities:
1 - 4x > 7 and 1 - 4x < -7
Solving the first inequality:
1 - 4x > 7
Subtract 1 from both sides:
-4x > 6
Divide both sides by -4 (remembering to reverse the inequality because we are dividing by a negative number):
x < -1.5
Solving the second inequality:
1 - 4x < -7
Subtract 1 from both sides:
-4x < -8
Divide both sides by -4 (again, remember to reverse the inequality):
x > 2
Therefore, the solution to the inequality |1 - 4x| > 7 can be written as:
{x | x < -1.5 or x > 2}
Alternatively, we can write the solution as:
{x | -1.5 < x < 2}
To solve the inequality |1 - 4x| > 7, we need to consider two cases: when 1 - 4x is positive and when it is negative.
1. Case 1: 1 - 4x > 0
When 1 - 4x is positive, the absolute value |1 - 4x| becomes 1 - 4x, and the inequality becomes:
1 - 4x > 7
To solve this inequality, you can isolate the variable x by subtracting 1 from both sides and dividing by -4:
-4x > 6
x < -1.5
So the solution in this case is x < -1.5.
2. Case 2: 1 - 4x < 0
When 1 - 4x is negative, the absolute value |1 - 4x| becomes -(1 - 4x), and the inequality becomes:
-(1 - 4x) > 7
To solve this inequality, you need to reverse the inequality sign when multiplying or dividing by a negative number. In this case, we multiply both sides by -1:
1 - 4x < -7
Next, isolate the variable x by subtracting 1 from both sides and dividing by -4:
-4x < -8
x > 2
So the solution in this case is x > 2.
Since we are dealing with absolute values, we need to consider the union of both cases in the solution. Therefore, the final solution is:
{x | x < -1.5 or x > 2}