Introduction
Inequalities are fundamental in mathematics, helping us understand relationships between numbers and expressions. Today, we will explore the inequality (4x + 7 > x  13), graph its solutions, and analyze what this means in a realworld context.
Step 1: Solve the Inequality
First, let's manipulate the inequality to isolate (x).
Starting from: [ 4x + 7 > x  13 ]
Rearranging the Inequality

Add (4x) to both sides: [ 7 > 5x  13 ]

Add (13) to both sides: [ 20 > 5x ]

Divide both sides by (5): [ 4 > x \quad \text{or} \quad x < 4 ]
Thus, we find that the solution to the inequality is (x < 4).
Step 2: Graphing the Inequality
To visualize this inequality, we can graph the line represented by the equation (y = 4x + 7) and the line represented by (y = x  13).
1. Find Intersection Point
To graph these, it helps to find their intersection point by solving: [ 4x + 7 = x  13 ]
Rearranging gives: [ 7 + 13 = 5x \implies 20 = 5x \implies x = 4 ]
2. Coordinates of Intersection
At (x = 4): [ y = 4(4) + 7 = 16 + 7 = 9 ]
So the intersection point is ((4, 9)).
3. Graphing the Lines
 The line (y = 4x + 7) has a yintercept of (7) and a slope of (4).
 The line (y = x  13) has a yintercept of (13) and a slope of (1).
4. Shading the Inequality
Since our solution states (x < 4), we will shade the area to the left of the vertical line at (x = 4). Additionally, since the inequality is strict (greater than, but not equal to), we will use a dashed line to indicate that (x = 4) is not included in the solution set.
Step 3: Analyzing the Result
The solution (x < 4) indicates that all values of (x) less than (4) will satisfy the original inequality.
Practical Example
Let's say (x) represents the age of children attending a summer camp, and the camp allows children less than (4) years old. Therefore, children aged (3) years and below qualify for the camp.
Conclusion
Understanding inequalities is crucial in mathematics and realworld applications. By solving and graphing (4x + 7 > x  13), we see that the solution is all values of (x) less than (4), highlighted graphically with a dashed line to denote that (4) itself is not included in the solution set.
For further exploration, practice solving similar inequalities, and consider how the graphical representation aids in understanding the solutions visually.
References
The methodology and framework for solving inequalities were inspired by inquiries and solutions from the BrainlY community, where users collaborate to enhance their understanding of mathematical concepts.