Problem 45 - Algebra 2

Question

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{y \geq-2 x+4} \\ {x>-3} \\ {y \geq 1}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the region where all shaded areas overlap. It is bounded by the line \(y = -2x + 4\), the vertical line \(x = -3\), and the horizontal line \(y = 1\).

Step 1: Graph the inequality \(y \$ -2x + 4\)

First, graph the line \(y = -2x + 4\) as if it were an equation. Use a solid line because the inequality includes equality (\(y \geq -2x + 4\)). Then, shade above the line because \(y\) is greater than or equal to the expression.

Step 2: Graph the inequality \(x > -3\)

Graph the vertical line \(x = -3\) using a dashed line because the inequality does not include equality (\(x > -3\)). Shade to the right of the line because \(x\) is greater than -3.

Step 3: Graph the inequality \(y \geq 1\)

Graph the horizontal line \(y = 1\) as a solid line because the inequality includes equality (\(y \geq 1\)). Then, shade above the line because \(y\) is greater than or equal to 1.

Step 4: Find the solution region

The solution to the system of inequalities is the region where all shaded areas overlap. Identify this region on the graph. It is the area above or on the line \(y = -2x + 4\), to the right of the line \(x = -3\), and above or on the line \(y = 1\).

Key Concepts

Inequality Graphing

Inequality graphing is a visual way to represent solutions of inequalities. Unlike equations that depict a single line, inequalities often involve a shaded region that includes a multitude of potential solutions. To graph an inequality, first consider it as an equation to find the boundary.

When dealing with a linear inequality, such as \(y \geq -2x + 4\), you begin by drawing the line \(y = -2x + 4\) as if it were a standard linear equation. The boundary line helps separate the graph into two regions. An important aspect when sketching the boundary is the type of line to use; a solid line indicates that points on the line are part of the solution set (as in \(\geq\) or \(\leq\)), while a dashed line indicates they are not (as in \(>\) or \(<\)).

Shading is a crucial element in inequality graphing. It represents all the possible solutions that satisfy the inequality. For the given inequality, the region above the line \(y = -2x + 4\) is shaded since it includes all points for which the value of \(y\) is greater than or equal to \(-2x + 4\). Thus, by combining the correctly drawn line with the proper shaded area, you effectively graph the inequality.

Linear Inequalities

Linear inequalities, such as \(y \geq 1\), resemble linear equations, but instead of one exact solution, they represent a range of solutions forming an area on a graph. For instance, graphing \(y \geq 1\) requires drawing a horizontal line at \(y = 1\). This line serves as the cutoff; any points above it meet the condition of \(y\) being greater than or equal to one.

Distinguishing Boundary Lines

The type of line used impacts the solutions included in the inequality. In the given example, because it is \(\geq\), the line is solid, suggesting that points lying exactly on \(y = 1\) are valid solutions. However, if the inequality were \(y > 1\), the line would be dashed, excluding solutions that fall precisely on the line itself.

To provide comprehensive solutions and optimizations in educational content, it is critical to highlight nuances such as the significance of line types in graphing linear inequalities, ensuring that students fully grasp the reasoning behind the graphical representations.

System of Inequalities

A system of inequalities involves multiple inequalities that must be simultaneously satisfied. Solving such a system by graphing means finding the common shaded area that represents where all inequalities overlap. This solution region depicts all the ordered pairs that satisfy every inequality in the system.

When addressing the system of inequalities given in the original exercise, you deal with three different inequalities that have their own boundary lines and shading directions:

• \(y \geq -2x + 4\) – Solid line, shade upwards.
• \(x > -3\) – Dashed line, shade to the right.
• \(y \geq 1\) – Solid line, shade upwards.

By graphing these inequalities on the same coordinate plane, students must identify the region where all the shadings intersect. This common area is the graphical solution to the system. It represents every possible pair of \(x\) and \(y\) values that satisfy all the inequalities at once. Understanding and identifying this region is key to mastering the concept of systems of inequalities.