- The boundary line is \(u = 2x - 5\), which has a slope \(m = 2\) and a \(u\)-intercept \(b = -5\).
- Draw this as a dashed line to indicate the strict inequality.
- Test the point \((0,0)\) to find that the region above the line satisfies the inequality.
- Shade the half-plane above the dashed line.
- The final values are: slope = \(\boxed{2}\), \(y\)-intercept = \(\boxed{-5}\).
Explanation
- Identifying the Slope and Intercept
The inequality is given by \(u > 2x - 5\). This is a linear inequality in two variables, \(x\) and \(u\). The boundary of the solution set is defined by the linear equation \(u = 2x - 5\). This equation is in the slope-intercept form \(u = mx + b\), where \(m\) is the slope and \(b\) is the \(u\)-intercept (the value of \(u\) when \(x = 0\)). By comparing \(u = 2x - 5\) with \(u = mx + b\), we can identify the slope \(m = 2\) and the intercept \(b = -5\).
- Graphing the Solution Set
To graph the solution set, we first draw the boundary line \(u = 2x - 5\). Because the inequality is strict (\(>\)), we must draw the line as a dashed line, indicating that points on the line are not included in the solution set. We can plot the line by finding two points, such as \(x = 0, u = -5\) and \(x = 2.5, u = 0\). Once the dashed line is drawn, we determine which region to shade by testing a point that is not on the line. Choosing the origin \((0, 0)\), we substitute its coordinates into the original inequality: \(0 > 2(0) - 5\), which simplifies to \(0 > -5\). Since this statement is true, the region containing the origin must be shaded. This corresponds to the half-plane above the line.
Examples
In urban planning, architects often use linear inequalities to designate zones for development. For example, if a developer needs the area of a commercial park (let \(x\) represent the land area allocated for parking and \(u\) represent the area for buildings) to be greater than a specific constraint relative to the parking capacity, they might model it as \(u > 2x - 5\). By visualizing this on a graph, they can clearly see all the viable combinations of parking and building space that meet their regulatory requirements, ensuring both efficiency and compliance with zoning laws.