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Magazine Chapter 6: Problem 15
$$ \text { Graph } y \leq-2(x-5) \text {. } $$
Short Answer
Expert verified
Graph the line \( y = -2x + 10 \), use a solid line, and shade below it.
Step by step solution
01
Identify the Equation Form
The given inequality is \( y \leq -2(x-5) \). This is an inequality in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, the expression must be rearranged to exactly match this form if desired.
02
Simplify the Expression
Distribute the \(-2\) across the \((x-5)\) to simplify it: \( y \leq -2x + 10 \). Now, \( y = -2x + 10 \) represents the boundary line of the inequality.
03
Graph the Boundary Line
Draw the line \( y = -2x + 10 \) on the coordinate plane. Begin by plotting the y-intercept (0,10). Use the slope \(-2\) to find another point: From (0,10), move down 2 units and right 1 unit to plot (1,8). Connect these points with a solid line because the inequality includes \( y = -2x + 10 \) (denoted by '鈮�').
04
Determine the Shading Region
The inequality is \( y \leq -2x + 10 \), so you will shade below the line. This indicates all the solutions where \( y \) is less than or equal to the line's value at a given \( x \).
05
Verify a Solution Point
Pick a test point not on the line (such as (0,0)) to check if it satisfies the inequality. Substitute into the inequality: \( 0 \leq -2(0) + 10 \rightarrow 0 \leq 10 \), which is true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Inequalities
Graphing inequalities is a visual way to represent solutions of inequalities on a coordinate plane. It involves both the graph of a boundary line and the shading of a region. To successfully graph an inequality:
- Identify if the inequality sign is 'greater than' (>), 'less than' (<), 'greater than or equal to' (鈮�), or 'less than or equal to' (鈮�).
- Draw the boundary line, based on the equation form of the inequality. If the inequality includes equality (like 鈮� or 鈮�), the boundary line is drawn as a solid line. If the inequality is strict (< or >), use a dashed line.
- Determine which part of the graph to shade, representing all the possible solutions of the inequality. For 鈮� and <, shade below the line; for 鈮� and >, shade above.
- Verify your solution by selecting a test point not on the boundary line to ensure it satisfies the inequality.
Slope-Intercept Form
The slope-intercept form of a linear equation is essential for graphing because it provides clear information about the line. It follows the format: \( y = mx + b \).
- \( m \) represents the slope of the line, which tells you the steepness and direction of the line. A positive slope means the line inclines upwards, while a negative slope means it declines.
- \( b \) is the y-intercept, indicating where the line crosses the y-axis. This is a key starting point when plotting the graph.
Coordinate Plane
The coordinate plane is a two-dimensional space where points are defined by pairs \( (x, y) \). It's a crucial tool for graphing everything from lines to complex functions and inequalities.
- The horizontal axis is the x-axis, while the vertical axis is the y-axis. Each point is placed based on its x (horizontal) and y (vertical) values.
- The intersection of the x and y axes is the origin, denoted as (0,0), providing a central reference point.
- Graph regions are divided into four quadrants. Each quadrant has different signs for x and y values. Quadrant I has positive values for both x and y.
Boundary Line
The boundary line in graphing linear inequalities is the line that represents the equation portion of the inequality.
- It serves as the limit separating the solution region from the non-solution region on the coordinate plane.
- To determine the nature of the boundary line, observe the inequality sign. '鈮�' and '鈮�' involve solid lines, indicating the line itself is part of the solution set. '<' and '>' involve dashed lines, signaling the line is not part of the solution set.
- Once the line is drawn, use it to help determine which side of the line to shade, showing all points that satisfy the inequality.
