Problem 12 - Elementary Algebra Within Reach

Question

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} y<-x+3 \\ y \geq-1 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution of the system of inequalities is the region below the line \(y = -x + 3\) and on or above the line \(y = -1\), where the shaded regions from both inequalities overlap.

Step 1 - Plot the lines

The first line is represented by \(y = -x + 3\), which has a slope of -1 and a y-intercept of 3. It can be drawn on a graph starting at the point (0, 3) and then moving down and to the right. The second line is represented by \(y = -1\) which is a flat line crossing the y-axis at -1.

Step 2 - Apply the inequalities

The first inequality was \(y < -x + 3\). This means that the valid y-values are those below the line. Therefore, shade everything below the line excluding the line itself, since the inequality does not have the or-equal-to component. The second inequality was \(y \geq -1\). This means that the valid y-values are those on and above the line y = -1. Therefore, shade everything on the line and above it.

Step 3 - Find the overlapping region

The solution to this system of inequalities is the area where the shading from both inequalities overlaps. This region represents all the points \((x, y)\) that satisfy both inequalities simultaneously.

Key Concepts

System of Inequalities

A system of inequalities involves multiple inequalities that must all be satisfied at the same time. Unlike equations, inequalities indicate a range of potential solutions. In this exercise, you have two inequalities:

• \(y < -x + 3\): This inequality represents all the points below the line \(y = -x + 3\).
• \(y \geq -1\): This inequality includes all the points on and above the line \(y = -1\).

The goal is to find a region on the graph where both conditions are true, which means both inequalities hold simultaneously for the same set of coordinates \((x, y)\). Understanding how to work with systems of inequalities is essential for solving many real-life problems where constraints must be met concurrently.

Plotting Lines

Plotting lines forms the foundation of graphing linear inequalities. Start by converting each inequality into an equation:

• For \(y = -x + 3\), plot the y-intercept at \((0, 3)\), then use the slope to determine the direction and angle of the line. A slope of -1 indicates a downward slant from left to right.
• For \(y = -1\), simply draw a horizontal line intersecting the y-axis at -1.

By plotting these lines on a coordinate plane, you create the boundaries that will help visualize the regions defined by these inequalities. This is a crucial step in visually assessing where solutions to the system of inequalities exist.

Overlapping Regions

A key concept when working with systems of inequalities is finding the overlapping regions. After plotting both lines, examine where the shaded areas from each inequality intersect. This overlap is the solution set for the system. Imagine the graph as a piece of paper with various sections highlighted. The overlapping highlighted area is where all conditions specified by the inequalities are satisfied. In this problem:

• The intersection lies below the line \(y = -x + 3\) and on or above the line \(y = -1\).

This region visually represents all the solutions satisfying both inequalities. Accurately determining this area is crucial as it directly influences the answer to the exercise.

Slope and Intercept

Understanding slope and intercept is essential for graphing linear equations and inequalities. The slope of a line indicates its steepness and direction:

• A slope of -1 in the line \(y = -x + 3\) means the line descends one unit vertically for every unit it moves horizontally.

Intercepts provide starting points:

• The y-intercept of 3 for \(y = -x + 3\) is where the line crosses the y-axis.
• The line \(y = -1\) intercepts the y-axis at the point \((0, -1)\), illustrating a horizontal line.

Recognizing these elements aids in sketching the correct representation of equations and understanding the geometry of the graph.

Shading Regions

Shading is integral when working with inequalities. It helps identify valid solution areas:

• For \(y < -x + 3\), shade below the line. The line itself is not included because it uses \(<\), indicating 'less than' without equality.
• For \(y \geq -1\), shade on and above the line. The \(\geq\) symbol means 'greater than or equal to,' so the line itself is part of the solution.

Shading visually separates the graph, highlighting areas that satisfy each inequality. The overlap from these shaded regions reflects where both conditions in the system of inequalities are true, providing an immediate, comprehensible way to view and determine solutions.