Problem 74 - Algebra 1

Question

Sketch the graph of the inequality in a coordinate plane. $$ y \leq-1.5 $$

Step-by-Step Solution

Answer

The graph includes a line on \(y = -1.5\) and shading below the line, indicating all the points where \(y \leq -1.5\).

Step 1: Understand the inequality

The inequality \(y \leq -1.5\) means that we want to find all the (x,y) pairs such that the y-coordinate is less than or equal to -1.5. This corresponds to a horizontal line along the y = -1.5 and all the area below it.

Step 2: Draw the line

Next, draw a horizontal line on your coordinate plane at where y equals -1.5. This line includes every x-coordinate because we don't have restrictions on x in our inequality.

Step 3: Shade the area below

The inequality specifies \(y \leq -1.5\), meaning the line and the area below it (where y is less than -1.5) should be shaded to indicate the solution area.

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is a two-dimensional surface that allows us to visualize mathematical problems and solutions. It consists of two number lines, called axes, that intersect perpendicularly.
The horizontal axis is known as the x-axis, while the vertical one is the y-axis. These axes divide the plane into four quadrants. The point where they intersect is the origin, denoted as \((0,0)\).

• The x-axis shows the horizontal direction.
• The y-axis shows the vertical direction.

When graphing inequalities, the coordinate plane helps us represent all possible solutions visually. By plotting equations and inequalities, you can quickly see which regions of the plane contain the solutions you're looking for.

Inequality Solutions on the Graph

Inequality solutions are quite different from equations, as they look for a range of values rather than a specific set of values. For the inequality \(y \leq -1.5\), we want to find all points where the y-coordinate is less than or equal to \(-1.5\). Here’s how it works: - **Solid Line:** Because the inequality is \("\leq"\), the line \(y = -1.5\) is included in the solution set and drawn solid. Any point on this line satisfies the equality part of the inequality.- **Solution Area:** Below the line is shaded to represent every point where the y-value is less than \(-1.5\). These combined make up the complete solution set.Drawing inequality solutions helps in easily seeing which parts of the coordinate plane satisfy the inequality, making critical mathematical analysis much simpler.

Recognizing a Horizontal Line

A horizontal line is a straight line that runs left to right and remains constant in y-value. With the equation \(y = -1.5\), this explains why the horizontal line is drawn at \(y = -1.5\) across the entire x-axis.
Key features include:

• The slope of a horizontal line is zero because the y-value doesn’t change with different x-values.
• It stretches indefinitely left and right, covering all x-coordinate values.

When graphing, it’s crucial to distinguish these characteristics to correctly identify and represent such lines. A clear understanding assists in interpreting problems and solutions that involve horizontal lines on the coordinate plane.

Graph Shading for Inequalities

Shading is a method in graphing to visually indicate the area that satisfies a given inequality. For \(y \leq -1.5\), shade below the line \(y = -1.5\) because this area contains all y-values less than \(-1.5\).
Steps for shading:

• Identify whether the area should be above or below the line based on the inequality sign ("\(\leq\)" means below, "\(\geq\)" means above).
• Make sure the boundary line (here, \(y = -1.5\)) is solid, as the inequality is inclusive of the line.
• Use consistent shading throughout to clearly show where the solutions lie.

Graph shading not only highlights the solution region but also enhances the visualization process of solving inequalities. It confirms at a glance if a point (\(x, y\)) is a solution or not.