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Linear inequalities are mathematical expressions that relate a linear function through inequality symbols such as <, >, ≤, or ≥. They describe a range of possible values rather than one specific solution. In everyday terms, you can think of them as statements that say "one quantity is larger or smaller than another." An inequality like \( y > -7 \) indicates that we are looking at all values of \( y \) that are greater than -7. It means the solutions are part of a set rather than a single point, making them crucial in understanding ranges and constraints in real-world problems.
Linear inequalities don't often include an \( x \)-term, as is the case in our example, but when they do, they can form lines of varying slopes. In simpler cases like \( y > -7 \), the graph becomes a horizontal line that helps visualize all potential values of \( y \).

Horizontal lines in graphing are straight lines that run parallel to the \( x \)-axis. They indicate that across all values of \( x \), \( y \) has a constant value. In the context of inequalities such as \( y > -7 \), the horizontal line is drawn at the point \( y = -7 \). This placement visually represents where the boundary of the solution set is located.
Understanding horizontal lines is critical as it shows that no matter what \( x \) is, \( y \) remains the same at the line. This is particularly useful in inequalities because it paints a distinct boundary of limitations that \( y \) must exceed or fall short of. The horizontal line at \( y = -7 \) offers a clear visual point separating values that satisfy the inequality from those that do not.

A dotted line is used when graphing inequalities like \( y > -7 \) to indicate that the line itself is not part of the solution set. It signals to viewers that values exactly at \( y = -7 \) do not satisfy the inequality; only values above it count.
The choice of a dotted line, as opposed to a solid one, conveys exclusion. By drawing a dotted line, you visually express that there's a boundary — a limit that values are close to but don't include. This aids in the interpretation of inequalities greatly, preventing confusion over whether a line value should be part of the solution.

Solution set shading is the technique used to graphically show all possible solutions of an inequality on a graph. In the case of \( y > -7 \), the solution set consists of all points above the line \( y = -7 \). Shading the area above this line helps identify which region of the graph represents valid \( y \) values.
Here’s how to effectively shade the solution:

• Identify the boundary line, as explained earlier.
• Note the direction indicated by the inequality symbol (greater than, in this case).
• Shade the region that correspondingly represents all values above the line.

Shading clarifies the regions where the inequality holds true, making it easier to visually confirm or explore further problems based on these solutions. It serves as an effective visual tool that communicates both boundaries and possibilities within a graph.