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Linear inequalities are expressions that involve a linear equation with an inequality sign (<, ≤, >, or ≥) instead of an equals sign. They represent regions of the graph where certain conditions are met. For example, the inequality \( y < -1 \) indicates that we are looking at the area on the graph where all the y-values are less than -1.

When dealing with linear inequalities, you’ll typically follow these steps:

• Understand the inequality and determine its meaning.
• Graph the associated boundary line.
• Shade the appropriate region (above or below the line).
• Use a test point to verify the correct region.

Linear inequalities are useful in various applications, such as economics, engineering, and everyday problem-solving scenarios.

The boundary line in a linear inequality is essentially the line you would get if you replaced the inequality sign with an equals sign. For the inequality \( y < -1 \), the boundary line is \( y = -1 \). This line acts as a border separating the solution region from the non-solution region.

If the inequality is strict (using < or >), you use a dashed line to indicate that points on the line are not included in the solution set. For example, in \( y < -1 \), the boundary line is dashed because points exactly at \( y = -1 \) are not part of the solution. If the inequality is non-strict (using ≤ or ≥), a solid line is used.

The steps to graph the boundary line are:

• Plot points that satisfy the equation of the boundary line.
• Draw the line (solid or dashed as appropriate).

For our exercise, plot \( y = -1 \) as a horizontal dashed line crossing the y-axis at -1.

The shaded region on a graph of a linear inequality represents all the possible solutions to the inequality. In the case of \( y < -1 \), the shading will be below the boundary line since we want all y-values less than -1.

The process includes:

• Determining which side of the boundary line to shade by using a test point.
• Visualizing the inequality to understand which region contains the solutions.

Shading helps to make it clear which part of the graph satisfies the inequality. For instance, after identifying that \( (0, 0) \) does not satisfy \( y < -1 \) (since 0 is not less than -1), the shading will be done below the dashed line.

A test point is a point not on the boundary line, used to determine which region of the graph to shade. A common choice is the origin, \( (0, 0) \), because it usually simplifies calculations.

Here’s how you use a test point:

• Choose a point not on the boundary line (e.g., \( (0, 0) \)).
• Substitute the coordinates of the test point into the inequality.
• If the inequality is true, the region containing the test point is part of the solution set. If false, shade the other side.

In the exercise’s case, using the point \( (0, 0) \) in the inequality \( y < -1 \) gives \( 0 < -1 \), which is false. Therefore, the region below \( y = -1 \) is shaded. This ensures that all y-values in that region are less than -1, confirming the solution.