Problem 19 - Thinking Mathematically

Question

Graph each linear inequality. \(y>-4\)

Step-by-Step Solution

Answer

To graph the inequality \(y > -4\), a horizontal line is placed at \(y = -4\), and the area is shaded above that line to represent the solution set. Any point in the shaded area satisfies the inequality.

Step 1: Plot the Line

Start by drawing a horizontal line on the y-axis at the point y = -4. Since the original inequality is stated as 'greater than' and not 'greater than or equal to', the line is dotted to indicate that points on the line itself are not included in the solution set.

Step 2: Indicate the Solution Set

Next, the area representing the solution set needs to be shaded. Since the inequality is 'y is greater than -4', the shading is done above the line. Every point in the shaded area has a y-coordinate greater than -4, so it is part of the solution set.

Step 3: Verification

Choose a random point in the shaded region, say (0, 0). Verify whether or not this satisfies the inequality. As 0 > -4, the chosen point is part of the solution set, confirming the answer is correct.

Key Concepts

Graphing Inequalities

Graphing inequalities involves a few straightforward steps that visually represent a range of solutions on a coordinate plane. When you have an inequality in the form of a linear equation, such as \(y > -4\), you're tasked with showing all the potential solutions that satisfy that condition. You start with drawing the boundary line, which separates the solution area from the non-solution area. If the inequality is "greater than" or "less than" (using symbols \( > \) or \( < \)), then you draw a dotted line. For "greater than or equal to" or "less than or equal to" (\( \geq \) or \( \leq \)), you use a solid line. The boundary describes the threshold of solutions, and the next step is shading the specific region to represent all possible solutions for the inequality.

Solution Set

The solution set of an inequality is all of the values that satisfy the equation. For \(y > -4\), every point in the coordinate plane where the y-coordinate is greater than -4 forms the solution set. Once the line is plotted, shading is crucial to visually capture this set. For "greater than" inequalities, the area above the line is shaded, signaling all y-values above and not including the line. Conversely, for "less than" inequalities, shading appears below the line. For validation, you can choose any point within the shaded area. Check if this point satisfies the inequality, reinforcing that it belongs to the solution set.

Coordinate Plane

The coordinate plane is a two-dimensional number line where each point is defined by a pair of numerical coordinates. These coordinates, typically represented as \((x,y)\), show horizontal and vertical positions. In graphing inequalities, the plane is the canvas where inequalities are visually expressed. The x-axis is horizontal, and the y-axis is vertical, intersecting at the origin, \((0,0)\). For our inequality, \(y > -4\), we only consider the y-axis to determine the boundary line at \(y = -4\). Once the line is drawn, you plot the entire solution region accordingly, using the coordinate axes as a guide.

Dotted Line

A dotted line is crucial when graphing certain inequalities. It serves as a boundary that does not include the points along the line itself. In the example \(y > -4\), the dotted line at \(y = -4\) indicates that the solutions are strictly greater than -4. This distinction is important: it separates "exclusive" conditions from "inclusive". If an inequality were \(y \geq -4\), a solid line would be used to include the values on the line as part of the solution. Therefore, the type of line directly represents whether or not the boundary line's points are considered solutions. This visual cue is essential in graphing to ensure accurate communication in mathematical representation.