Problem 49 - Elementary Algebra

Question

$$ \text { For Problems 37-56, graph each linear inequality. (Objective } 3 \text { ) } $$ $$ -x+2 y<-2 $$

Step-by-Step Solution

Answer

Shade the region below the line \(y = \frac{1}{2}x - 1\).

Step 1: Rewrite the Inequality as an Equation

First, we need to find the line that represents the boundary for the inequality. Rewrite the inequality \(-x + 2y < -2\) as an equation:\(-x + 2y = -2\).

Step 2: Solve for y

Solve the equation for \(y\) to make graphing easier:\[-x + 2y = -2\]Add \(x\) to both sides:\[2y = x - 2\]Divide every part by 2 to solve for \(y\):\[y = \frac{1}{2}x - 1\].

Step 3: Graph the Line

Plot the line \(y = \frac{1}{2}x - 1\). Start by marking the y-intercept \((0, -1)\) on the graph. Then, using the slope \(\frac{1}{2}\), rise up 1 unit and run 2 units to the right to find another point \((2, 0)\). Connect these points with a dashed line since the inequality is \('<'\), not \('≤'\).

Step 4: Determine the Shaded Region

Choose a test point not on the line, like \((0, 0)\). Substitute it into the original inequality to determine which side of the line should be shaded:\[-(0) + 2(0) < -2\]This simplifies to \(0 < -2\), which is false, so the region that does not contain \((0, 0)\) should be shaded.

Step 5: Finalize the Graph

The solution to the inequality is the region below the dashed line \(y = \frac{1}{2}x - 1\). Make sure this region is clearly shaded on the graph to show all solutions to the inequality.

Key Concepts

Linear Equation

A linear equation is any equation that can be written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Linear equations represent straight lines on a graph. You can think of them as recipes for creating lines. Each point on the line is a solution to the equation, meaning it satisfies the equation when you plug in the \(x\) and \(y\) values.The concept is simple: if you know what two points on a line are, you can draw the entire line just by connecting those points. Linear equations are foundational in graphing because they help create and understand the relationships between variables.With practice, spotting and transforming equations into their graphical representations will become second nature. Linear equations lay the groundwork for understanding more complex mathematical concepts.

Slope-Intercept Form

Slope-intercept form is a way of writing linear equations so they are easy to graph. The typical format is \(y = mx + b\), where:

• \(m\) is the slope of the line.
• \(b\) is the \(y\)-intercept of the line.

The slope \(m\) determines how steep the line is and the direction it goes. Essentially, the slope tells you how much \(y\) changes for a change in \(x\).The \(y\)-intercept \(b\) is the point where the line crosses the \(y\)-axis. This is a starting point for graphing the line. Using the slope-intercept form makes it much easier to visualize the line and quickly start graphing it on the coordinate plane. Just plot the \(y\)-intercept, apply the slope, and you're good to go!

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can graph equations. It's like a map, with an \(x\)-axis (horizontal) and a \(y\)-axis (vertical). These axes divide the plane into four quadrants.Each point on the coordinate plane is defined by an \(x\) and \(y\) coordinate. The point \((x, y)\) tells you how far to move horizontally (right or left) and vertically (up or down) from the origin, which is the point where the axes intersect at \((0,0)\).Graphing on the coordinate plane helps in visualizing mathematical relationships and solving equations. It's a powerful tool in math that lets you see what's happening in an equation, not just calculate it.

Inequality Shading

Inequality shading is used when graphing inequalities rather than equations. In inequalities, solutions are often not just one line but a whole region of the coordinate plane.To determine which part of the plane to shade, first graph the associated linear equation as a boundary. For < or > inequalities, use a dashed line to show the boundary line itself is not part of the solution. For ≤ or ≥, use a solid line.Then, pick a test point not on the line, like \(0, 0\), and substitute it into the inequality:

• If it's true, shade the side containing this point.
• If false, shade the opposite side.

Shading indicates all the possible solutions for the inequality, visually representing where one variable is smaller or larger than the other based on the linear relationship.