Graph the solution to this inequality on the number line. -4x+7>x-13
The Correct Answer and Explanation is :
To solve the inequality (-4x + 7 > x – 13), we start by isolating the variable (x) on one side of the inequality.
Step 1: Rearrange the inequality
We can move all terms involving (x) to one side and constant terms to the other. First, we subtract (x) from both sides:
[
-4x – x + 7 > -13
]
This simplifies to:
[
-5x + 7 > -13
]
Next, we subtract (7) from both sides:
[
-5x > -13 – 7
]
This simplifies to:
[
-5x > -20
]
Step 2: Divide by (-5)
Now, we divide both sides of the inequality by (-5). Remember that when we divide or multiply by a negative number, we need to flip the inequality sign:
[
x < \frac{-20}{-5}
]
This simplifies to:
[
x < 4
]
Step 3: Graph the solution on a number line
To graph the solution (x < 4) on a number line:
- Draw a horizontal line (the number line).
- Mark the point (4) on the line.
- Since the inequality is strict ((<)), we use an open circle at (4) to indicate that (4) is not included in the solution.
- Shade the line to the left of (4) to indicate all numbers less than (4) are included in the solution.
Explanation
The solution (x < 4) means that any number less than (4) satisfies the inequality. In practical terms, this indicates a range of values that (x) can take. For instance, (3), (0), and (-1) would all be valid solutions, while (4) and any number greater than (4) would not satisfy the inequality. This concept is important in various applications, such as determining thresholds in real-world scenarios, optimization problems, and analyzing systems where specific constraints must be met.