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Magazine Chapter 3: Problem 251
Graph the linear inequality: \(y>\frac{2}{3} x-1\)
Short Answer
Expert verified
Plot points (0, -1) and (3, 1), draw a dashed line through them, and shade the region above the line.
Step by step solution
01
- Rewrite the Inequality in Slope-Intercept Form
The given inequality, \(y > \frac{2}{3} x - 1 \), is already in slope-intercept form, \(y = mx + b \), where the slope \(m = \frac{2}{3} \) and the y-intercept \( b = -1 \).
02
- Plot the Y-Intercept
Identify the y-intercept on the graph. This is where the line crosses the y-axis. For the inequality \(y > \frac{2}{3} x - 1 \), the y-intercept is \( b = -1 \). Plot this point at \( (0, -1) \).
03
- Use the Slope to Plot Another Point
Use the slope, \( \frac{2}{3} \), which indicates a rise of 2 units and a run of 3 units. From the intercept point \( (0, -1) \), move up 2 units and to the right 3 units to locate the next point: \( (3, 1) \). Plot this point.
04
- Draw the Boundary Line
Draw a dashed line through the points \( (0, -1) \) and \( (3, 1) \). Use a dashed line because the inequality symbol is '>', which means the line itself is not included in the solution set.
05
- Shade the Solution Region
Since the inequality is \( y > \frac{2}{3} x - 1 \), shade the region above the line, as this represents all the points where \( y \) is greater than \( \frac{2}{3} x - 1 \).
06
- Verify a Point
Verify that the shading is correct by choosing a test point not on the boundary line. For example, use the point \( (0, 0) \). Substitute into the inequality: \( 0 > \frac{2}{3} (0) - 1 \) simplifies to \( 0 > -1 \), which is true. This confirms that the region above the dashed line is the correct solution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
linear inequalities
Linear inequalities involve linear expressions that use inequality signs like '>', '<', '鈮�', or '鈮�' rather than an equals sign. They describe a range of possible values for variables, rather than a single value. For example, the inequality
\( y>\frac {2} {3} x-1 \) describes a set of \((x, y)\) pairs where \( y \) is greater than \(\frac {2} {3} x-1 \). Linear inequalities are useful in real-world situations where conditions are not exact, such as budgeting, and engineering tolerances.
\( y>\frac {2} {3} x-1 \) describes a set of \((x, y)\) pairs where \( y \) is greater than \(\frac {2} {3} x-1 \). Linear inequalities are useful in real-world situations where conditions are not exact, such as budgeting, and engineering tolerances.
slope-intercept form
The slope-intercept form of a linear equation is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is very useful for graphing because you can quickly identify two key features of the line:
- The \( y \)-intercept, which is where the line crosses the \( y \)-axis.
- The slope, which describes how steep the line is.
graphing inequalities
When graphing inequalities, the process is very similar to graphing equations, but with a few extra steps:
- First, graph the boundary line. If the inequality is strict (using '>' or '<'), use a dashed line to indicate the points on the line are not included.
- If it is inclusive (using '鈮�' or '鈮�'), use a solid line.
- Next, determine which side of the boundary line to shade.
- A good approach is to use a test point (like (0,0) if it is not on the line), substitute it into the inequality, and see if the inequality holds true.
mathematical graphing
Graphing is a powerful way to visualize equations and inequalities. Here are some tips to make your graphing process better:
- Use graph paper for more precise points and lines.
- Label your axes and include scale marks to help locate points accurately.
- Draw your axes with a ruler for straight lines.
boundary lines
Boundary lines are the lines that divide the graph into two regions. For linear inequalities:
- Dashed lines represent inequalities that are not inclusive ('\(>\)' or '\(<\)'), indicating the boundary itself is not part of the solution set.
- Solid lines are used for inclusive inequalities ('\(鈮)' or '\(鈮)'), meaning the points on the line are included in the solution set.
