To solve the inequality |4x – 1| < 5, we need to understand what the absolute value means. The expression |A| < B implies that -B < A < B. In this case, we can break it down into two separate inequalities:
- -5 < 4x - 1
- 4x – 1 < 5
Now, let’s solve each inequality one at a time.
1. Solve the first inequality: -5 < 4x - 1
Start by adding 1 to both sides:
-5 + 1 < 4x
-4 < 4x
Now, divide both sides by 4:
-1 < x
This simplifies to:
x > -1
2. Solve the second inequality: 4x – 1 < 5
Again, add 1 to both sides:
4x < 5 + 1
4x < 6
Now, divide both sides by 4:
x < 1.5
Combine the Results
From the two inequalities, we have:
- x > -1
- x < 1.5
We can combine these results into a single inequality:
-1 < x < 1.5
Graphing the Inequality
To graph the solution on a number line:
- Draw a number line.
- Mark the points -1 and 1.5.
- Use open circles at both -1 and 1.5 to indicate that these endpoints are not included in the solution.
- Shade the area between -1 and 1.5 to show that this range of values is the solution.
This gives you a visual representation of all the numbers that satisfy the inequality.