Problem 39 - Intermediate Algebra

Question

Graph each inequality. $$ x<-\frac{1}{2} y $$

Step-by-Step Solution

Answer

Graph the line \(y = -2x\) as dashed and shade above it.

Step 1: Rewrite the Inequality

The given inequality is \( x < -\frac{1}{2} y \). We can rewrite this inequality in a more familiar format, often used for graphing, such as \( y > -2x \) by solving for \( y \). First, multiply both sides by \(-2\) to solve for \( y \) (remember to flip the inequality sign).

Step 2: Interpret the Line Equation

The inequality \( y > -2x \) can be interpreted as describing all points that are above the line \( y = -2x \). This line has a slope of \(-2\) and passes through the origin \((0,0)\).

Step 3: Draw the Boundary Line

Draw the boundary line for \( y = -2x \) on a coordinate plane. Since the inequality is 'greater than' (\( > \)), use a dashed line to indicate that points on the line itself are not included in the solution set.

Step 4: Shade the Correct Side

Shade the region above the dashed line, \( y = -2x \), because the inequality sign is 'greater than' (\( > \)). This region represents the solution to the inequality, as it includes all points \((x, y)\) such that \( y > -2x \).

Key Concepts

Understanding Linear Inequalities

A linear inequality looks similar to a linear equation, but instead of an equals sign, it uses inequality symbols like "<", ">", "≤", or "≥". These symbols give us a range of possible solutions rather than a single line on the graph. For example, in the expression \( y > -2x \), the symbol \( > \) indicates that solutions are all points where the value of \( y \) is greater than twice the value of \( x \), but not equal to \( -2x \).
This inequality tells us about a wide array of possible solutions, compared to a precise solution when dealing with linear equations alone.

Navigating the Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and shapes. Each point on the plane is identified by a pair of numbers called coordinates, generally denoted as \((x, y)\). The horizontal line is known as the x-axis, and the vertical line is the y-axis.
To graph linear inequalities like \( y > -2x \) on the coordinate plane, we start by drawing the related line, \( y = -2x \). This serves as a boundary, helping us visualize which parts of the plane contain solutions. In this setup, the coordinate plane acts as the canvas where every spot can be tested against the inequality to see if it fits or not.

The Role of the Slope

The slope of a line tells us how steep the line is and in which direction it moves. It is often represented by the letter \( m \) in equations like \( y = mx + b \). For the inequality \( y > -2x \), the slope is \(-2\). This means that for every unit increase in \( x \), \( y \) decreases by 2 units.

• If the slope is positive, the line will rise to the right.
• If it's negative, like in our case, it will fall to the right.

Understanding the slope helps us accurately sketch the line and determine how to shade the areas representing solutions.

Identifying the Solution Set

The solution set for an inequality includes all the coordinate points that satisfy the inequality. For \( y > -2x \), the solution set comprises all the points above the boundary line \( y = -2x \).
When graphing, use a dashed line for \( y = -2x \) because the points on this line aren't part of the solution set. We only include points above this line, fulfilling the 'greater than' condition.

• Shade the area above the \( -2x \) line to depict this set visually.
• The shading visually separates the solutions from non-solutions, making it easier to see which x and y pairs are permissible.

By understanding solution sets, we effectively translate algebraic expressions into visual representations on the graph.