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The slope-intercept form is a linear equation of the form \(y = mx + b\). This formula is incredibly helpful when working with graphs, as it represents the equation of a straight line. Here, \(m\) stands for the slope of the line, describing how steep the line is. The slope is the ratio of the rise to the run between any two points on the line. - A positive \(m\) means the line rises as you move to the right. - A negative \(m\) indicates the line falls as you move to the right. The \(b\) denotes the y-intercept, which is where the line crosses the y-axis. An easy way to remember this is that the y-intercept is the value of \(y\) when \(x\) is zero. In our example, converting the inequality into slope-intercept form reveals \(y \ge -\frac{5}{3}x - 5\). This gives us a slope \(m = -\frac{5}{3}\) and y-intercept \(b = -5\), making graphing much simpler.

Linear inequalities are similar to linear equations but instead of equality, they express inequalities like greater than or less than. An inequality might be displayed as \(y > mx + b\) or \(y \le mx + b\). - Use a solid line for inequalities with \(\ge\) or \(\le\), indicating that points on the line satisfy the inequality. - Use a dashed line for inequalities with \(>\) or \(<\), as the line itself is not part of the solution.In our exercise, the inequality reads \(5x + 3y \ge -15\). Once transformed to slope-intercept form, it becomes \(y \ge -\frac{5}{3}x - 5\). This tells us that the area above and including the line is part of the solution. Using inequalities helps determine various conditions in real-world problems, like where a particular solution space exists.

Graph shading helps indicate which side of the line makes the inequality true. This is an essential step after plotting your line on a graph. To determine the correct region to shade: - Choose a test point that is not on the line, such as \((0, 0)\).- Substitute these coordinates into the inequality.- If the inequality holds true, shade the region containing the test point.In our example, substituting the test point \((0, 0)\) into the inequality \(5x + 3y \ge -15\) results in \(0 \ge -15\), which is true. Hence, we shade the region that includes this test point, indicating all solutions that satisfy the inequality. The shaded area graphically shows all possible solutions to the inequality, which can be especially informative when visualizing constraints in problems.