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When graphing inequalities like \( y \leq 1 \), the first step involves plotting a line that represents the boundary of the inequality. For the inequality \( y \leq 1 \), this means drawing a horizontal line at \( y = 1 \). This line is solid because the inequality symbol \( \leq \) includes equality, meaning points on this line are part of the solution set. When the inequality is \( y \leq 1 \), it indicates that all the solutions will be on or below the line. It is crucial to make the line clearly solid, distinguishing it from a dashed line used in strict inequalities like \( y < 1 \). A solid line confirms all the points on the line are included in the solution set. Graphing inequalities helps visualize all possible solutions and makes problem-solving or analysis more efficient. This visual method combines geometric concepts with algebraic equations, providing a comprehensive view of the inequality’s behavior.

In the context of inequalities, the solution set represents all the possible values that satisfy the inequality. For the inequality \( y \leq 1 \), the solution set includes every point on the line \( y = 1 \) and all the points below this line on the graph. Visualizing a solution set is vital as it shows not only where solutions exist, but also the extent of those solutions.

• Points on the line: These represent values where \( y = 1 \).
• Points below the line: These satisfy \( y < 1 \).

By depicting the solution set on a graph, learners can instantly see which values of \( y \) are valid for the inequality \( y \leq 1 \). This approach aids in understanding that an inequality encompasses a range of possible solutions, not just a single answer.

Shading regions on a graph is a vital part of solving inequalities visually. Once the boundary line \( y = 1 \) is drawn, the next step is to shade the region that represents the inequality's solutions. This type of shading indicates all the values that make the inequality true. For \( y \leq 1 \), shade the area below the line to cover all points satisfying the inequality. The shaded region visually represents the range of solutions and confirms any point within this area (including on the line itself) is a valid solution to \( y \leq 1 \).

• The entire area beneath the solid line is shaded to show inclusion of all points where \( y \leq 1 \).
• Shading is crucial as it differentiates between valid and invalid solutions at a glance.

This method is particularly helpful for complex systems of inequalities, where intersecting shaded areas show where multiple inequalities are met simultaneously.