Problem 45 - Elementary Algebra Within Reach

Question

In Exercises \(41-46\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{r} x \quad \geq 1 \\ x-2 y \leq 3 \\ 3 x+2 y \geq 9 \\ x+y \leq 6 \end{array}\right. $$

Step-by-Step Solution

Answer

A system of linear inequalities graph is composed of a multiple regions shaded to represent different inequalities. The overlap region represents the common solution to all inequalities. In this case, boundaries are the vertical line x=1, two sloping lines defined by the equations \(y = (x - 3) / 2\) and \(y = (9 - 3x) / 2\), and the line \(y = 6 - x\). The solution region is where all these areas overlap on the graph.

Step 1: Sketch the inequality \(x \geq 1\)

To do this, first graph the line \(x = 1\). This is a vertical line that cuts the x-axis at x=1. Next, because \(x\) is greater than or equal to 1, shade to the right side of the line.

Step 2: Sketch the inequality \(x - 2y \leq 3\)

Rearrange this inequality to \(y \geq (x - 3) / 2\). Draw the line corresponding to \(y = (x - 3) / 2\). Because \(y\) is greater than or equal to \((x - 3) / 2\), shade above the line.

Step 3: Sketch the inequality \(3x + 2y \geq 9\)

Rearrange this inequality to \(y \leq (9 - 3x) / 2\). Draw the line corresponding to \(y = (9 - 3x) / 2\). Because \(y\) is less than or equal to \((9 - 3x) / 2\), shade below the line.

Step 4: Sketch the inequality \(x + y \leq 6\)

Rearrange this inequality to \(y \leq 6 - x\). Draw the line corresponding to \(y = 6 - x\). Because \(y\) is less than or equal to \(6 - x\), shade below the line.

Step 5: Determine the overlapping region

Compare the four shaded regions. The overlapping region that satisfies all four inequalities is the solution to the system of inequalities.

Key Concepts

linear inequalities

Linear inequalities involve mathematical expressions where two expressions are related by inequalities such as \(\leq, \geq, <, >\). In our exercise, these inequalities include relationships of variables such as \(x \geq 1\), which means that \(x\) can be any number greater than or equal to 1. With inequalities, instead of finding specific values, we're looking for a range or set of solutions that satisfy the inequality condition. For multiple inequalities, the solutions must satisfy all given conditions in the system. This set of solutions typically forms a region on a graph.

graph sketching

Graph sketching is the process of plotting lines and curves of equations or inequalities on a coordinate plane. For graphing linear inequalities, the first step is to convert the inequality into an equation by replacing the inequality sign with an equality sign. This provides the boundary line for the inequality. For instance, with the inequality \(x - 2y \leq 3\), we first draw the line \(x - 2y = 3\). Adjusting for easy graphing, we rearrange it to \(y = \frac{x - 3}{2}\), which is a line with a slope and intercept that we can easily graph. Each inequality in our system gives a boundary that helps define the solution region.

shading regions

Once the line representing the boundary of an inequality is drawn, shading the regions is the next step to identify where the inequality holds.

• For \(x \geq 1\), we shade all the area to the right of the line, indicating all \(x\) values greater than or equal to 1.
• For \(y \geq \frac{x - 3}{2}\), the shading occurs above the line. This shows where \(y\) values meet or exceed the line's values.
• Similarly, for \(y \leq \frac{9 - 3x}{2}\) and \(y \leq 6 - x\), we shade below their respective lines.

The shaded area represents all solutions to that particular inequality. If the inequality is strict (e.g., \(<\) or \(>\)), the boundary line is often drawn as a dashed line, indicating that points on the line aren't included in the solutions, although our exercise uses "or equal to" signs, meaning solid lines are appropriate.

overlapping solution

The key to solving a system of linear inequalities is finding the common area where all inequalities' solutions overlap. This overlapping region represents the solution to the system, which satisfies each inequality simultaneously. By inspecting each shaded region from the inequalities:- The region to the right of \(x = 1\),- Above the line \(y = \frac{x-3}{2}\),- Below \(y = \frac{9-3x}{2}\),- And below \(y = 6-x\),we identify a polygonal region where all these conditions meet. This shared area is the solution set for the entire system. Graphically, ensure accuracy by re-checking intersections of boundary lines. Only within this common zone do all inequality conditions hold true, neatly illustrating how multiple conditions interact on a graph. This process of overlapping regions is essential in systems of inequalities, as it visually and mathematically illustrates the feasible solution set.