91Ó°ÊÓ

Graphing inequalities involves representing a range of solutions on a coordinate plane. Inequalities like the one in the exercise, \( F + 2S < 84 \), describe a set of conditions rather than a single value like equations do. To graph an inequality:

• First, determine the "boundary line" by converting the inequality to an equation. In this exercise, the boundary line is given by \( F + 2S = 84 \).
• Draw this line using two points. In this case, it connects the points where the line intersects the axes: \((S = 0, F = 84)\) and \((S = 42, F = 0)\).
• Shade the area below this line to represent all combinations where \( F + 2S < 84 \). A dashed line is appropriate here because the inequality is "less than," not "less than or equal to."

Graphing makes inequalities visually understandable, showing the range of possible solutions as a shaded region, indicating all of the combinations that satisfy the condition given by the inequality. This visual tool is invaluable in quickly identifying feasible solutions.

Linear equations are central in algebra and describe a straight line on a graph. They are equations of the first degree, meaning they have variables raised to a power of one. The basic form is often \( y = mx + b \), where:

• \(m\) represents the slope of the line.
• \(b\) represents the y-intercept.

In our exercise, the related linear equation is \( F + 2S = 84 \). This line serves as the boundary to the inequality. Finding points to graph this line involves setting one variable to zero and solving for the other:

• Set \( S = 0 \) to find \( F = 84 \).
• Set \( F = 0 \) to find \( S = 42 \).

Connecting these points provides the boundary line necessary to differentiate the solution set of the inequality. Working with linear equations helps students understand the constant relationship between variables.

The coordinate plane is a fundamental concept in graphing and visually representing algebraic solutions. It consists of two axes, typically labeled \(x\) (horizontal) and \(y\) (vertical). In the exercise, the horizontal axis represents the number of two-point shots \(S\), while the vertical axis represents one-point free throws \(F\).Understanding how to plot on a coordinate plane is essential:

• Each point is expressed as an ordered pair \((x, y)\) or here \((S, F)\).
• Points are positioned based on their values relative to the axes.

By accurately plotting points, such as the points \((S = 0, F = 84)\) and \((S = 42, F = 0)\), students can graph lines and inequalities. The coordinate plane serves as a backdrop to visualize mathematical relationships, making it easier to interpret solutions and comprehend the spatial aspect of algebra. This visualization helps students connect algebraic concepts with real-world applications.