Problem 13 - Elementary Algebra Within Reach

Question

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} y>2 x \\ y>-x+4 \end{array}\right. $$

Step-by-Step Solution

Answer

After graphing each inequality and shading the appropriate areas, the overlapping region above both lines, y=2x and y=-x+4, represents the solution to the system of inequalities.

Step 1: Graph the boundaries

Start by drawing the graphs of \(y=2x\) and \(y=-x+4\) as if they were equalities. Both graphs are straight lines. The first line passes through the origin (0,0) and has a slope of 2. The second line has a slope of -1 and y-intercept of 4.

Step 2: Shade the appropriate areas

The inequality \(y>2x\) means we're looking for all points (x, y) where y is greater than 2x. For each point above the line y=2x, y is greater than 2x, so shade the area above the line. Similarly, for \(y>-x+4\), we're looking for points where y is greater than -x+4. This corresponds to the area above the line y=-x+4, so shade that area as well.

Step 3: Identify the overlap

The solution to the system of inequalities is the set of points that satisfy both inequalities. This is represented by the region where the shaded areas from Step 2 overlap. Highlight this region on your graph.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting regions represented by inequalities on a coordinate plane. Unlike equations, which define a specific line or curve, inequalities define a range of values or areas above or below the line.

To graph an inequality, such as \( y > 2x \) or \( y > -x + 4 \), begin by treating the inequality sign as an equals sign. This means you first sketch the line \( y = 2x \) or \( y = -x + 4 \). This line is your boundary. The slope-intercept form makes it easy: \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.

Here’s how you proceed:

• Draw the boundary line. Solid line, if \( \ \geq \) or \( \ \leq \); dashed line for \( \ > \) or \( \ < \).
• Choose a test point not on the line, like \((0,0)\) if it's not on the boundary.
• Substitute the test point into the inequality. If true, shade the region containing the test point. If false, shade the opposite side.

The result is a visual representation of all solutions that satisfy the inequality, illustrating which side of the boundary your solutions lie.

Systems of Inequalities

A system of inequalities consists of two or more inequalities considered simultaneously. The solution is not just the solution to each individual inequality, but where their solutions overlap on a graph.

For the system: \[ \begin{cases} y > 2x \ y > -x + 4 \end{cases} \] The objective is to find a common region that satisfies all inequalities in the system.

Here's how you solve step-by-step:

• Graph each inequality separately using the method described in graphing linear inequalities.
• Look for the region where shaded areas overlap. Each inequality's graph will contribute to forming a boundary of this feasible region.
• This overlap represents the solution, which is all the points that meet all conditions simultaneously.

This approach can be used to find solutions in real-world optimization problems, where finding common ground is essential.

Algebraic Graphing Techniques

Algebraic graphing techniques provide tools for better understanding and visualizing math inequalities and equations. Key techniques include plotting points, understanding slope and intercept, and managing the visualization of inequalities.

When graphing lines, knowing the slope \( m \) and y-intercept \( b \) from \( y = mx + b \) makes it straightforward to plot them:

• The slope \( m \) dictates the line's steepness and direction.
• The y-intercept \( b \) tells where the line crosses the y-axis.

For example, in \( y = 2x \), the slope \( m = 2 \) tells us for every unit increase in \( x \), \( y \) increases by two units. Similarly, for \( y = -x + 4 \), the slope \( m = -1 \) implies a change in \( y \) in the opposite direction to \( x \).

Utilize dashed lines when the inequality does not include the boundary \(( \ > \ or \ < \)) and solid when it does \(( \ \geq \ or \ \leq \)). This distinguishes points that strictly fall within inequalities from those that also satisfy the boundary.

By mastering algebraic graphing techniques, you can create more accurate and meaningful visualizations, helping to solve inequalities and understand their implications in various contexts.