91Ó°ÊÓ

When dealing with inequalities like \( y > -3 \), determining the solution set is crucial. A solution set consists of all the permissible solutions that satisfy the inequality. For the problem at hand, the solution set includes every point on the coordinate plane where the \( y \) value is greater than \(-3\).

To visualize this, imagine a series of horizontal lines on the Cartesian plane where each line represents a possible \( y \) value. The solution set for \( y > -3 \) would then be all horizontal lines above the line \( y = -3 \). Remember, since \( y > -3 \) is a strict inequality, the line itself won’t be part of the solution set. This idea of "excluding" the boundary line from the solution gets translated into graphical representation using a dashed line, a concept we will delve into next.

The boundary line plays a fundamental role in graphing linear inequalities. It provides a visual marker of the values that come very close but do not satisfy the inequality when a strict inequality symbol is used.

In our example, the boundary line is \( y = -3 \). Since the inequality is \( y > -3 \), and does not include the equal sign \((\geq)\), the boundary line needs to be depicted as dashed. This communicates that points lying directly on \( y = -3 \) are not solutions to the inequality, a crucial aspect to grasp visually.

• Solid line: Used when variable equality is included (\( \geq \) or \( \leq \)).
• Dashed line: Used when the points on the line are not part of the solution (\( > \) or \( < \)).

Remember that accurately drawing and distinguishing this line helps convey whether points directly on the boundary are part of your solution set.

Shading is the technique used in graphing inequalities to exhibit all the points that do satisfy the inequality. It is both a visual cue and a solution representation for inequalities.

For \( y > -3 \), once the boundary line \( y = -3 \) is in place and dashed correctly, determining where to shade is the next step. Since we are concerned with points where \( y \) is greater than \(-3\), shading the area above the dashed line makes sense. This shaded region represents the solution set, showing that any point selected from this area will satisfy \( y > -3 \).

• Always look at the inequality sign to know where to shade:
• If \( y > ext{ or } y \geq \), shade above.
• If \( y < ext{ or } y \leq \), shade below.

Shading should be clear and distinct to help distinguish the solution region from non-solution areas. This visual distinction is key to correctly interpreting any graph of an inequality.