To solve the linear inequality y < 4x + 5, we will follow a series of steps to find the solution set.
1. Understanding the inequality:
The inequality states that for any point (x, y) that satisfies the condition, the y-value must be less than the value given by the equation of the line y = 4x + 5.
2. Graphing the boundary line:
The first step in visually understanding the inequality is to graph the associated boundary line. To do this, we need to convert the inequality.
Start by graphing the equation y = 4x + 5. This is the boundary line:
- The y-intercept is 5, meaning the line crosses the y-axis at (0, 5).
- The slope is 4, indicating that for every 1 unit increase in x, y increases by 4 units.
When you graph this line, make sure to use a dashed line instead of a solid line because the inequality sign (<) indicates that points on the line do not satisfy the inequality.
3. Choosing a test point:
Select a test point to determine which side of the line represents the solution set. A convenient test point is (0, 0) because it is easy to evaluate:
y < 4(0) + 5
0 < 5 (True)Since this is true, the area that includes (0, 0) is part of the solution set.
4. Shading the solution region:
Now, shade the region of the graph below the dashed line y = 4x + 5. This shading indicates that all points in this area satisfy the inequality y < 4x + 5.
5. Solution set:
The solution set for the inequality y < 4x + 5 includes all (x, y) coordinates that fall in the shaded area below the line. In interval notation, since we are focusing on the variable y, we can express the solution as:
{(x,y) | y < 4x + 5}This means all pairs (x, y) such that y is less than what is produced by the equation on the right side.
Summary:
To conclude, the solution to the linear inequality y < 4x + 5 can be represented graphically by shading the region below the dashed line on a graph, which visually depicts all the points that satisfy the inequality.