Problem 53 Sketch the graph of the inequali... [FREE SOLUTION]

Chapter 6: Problem 53

Sketch the graph of the inequality. $$x+6 y \leq 12$$

Short Answer

Expert verified

The graph of the inequality $x + 6y \leq 12$ is a line drawn through the point (0,2) with a negative slope of \frac{1}{6}. The region that satisfies the inequality (i.e., below the line or on the line) is shaded.

Step by step solution

Write the inequality as an equation

In order to graph the inequality $x + 6y \leq 12$, it is beneficial to first graph the linear equation $x + 6y = 12$. This equation is in the standard form of a linear equation, $Ax + By = C$. Rewrite the equation in the slope-intercept form, $y = mx + b$, to easily identify the slope and y-intercept. Doing this gives $y = -\frac{1}{6}x + 2$. Thus, the slope of the line is -\frac{1}{6} and the y-intercept is 2.

Plot the line

Draw the y-intercept on the graph at point (0,2). Since the slope is -\frac{1}{6}, for each unit increase in x, y will decrease by \frac{1}{6}. This calculated slope will help in drawing the line accurately on the graph.

Shade the region

The graph of the line $x + 6y = 12$ represents all points which satisfy the equation exactly. However, the original inequality was $x + 6y \leq 12$. This is satisfied for all points either on the line or below it. Thus, shade the region below the line to show this.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations

Linear equations are mathematical expressions that describe a straight line on a graph. These equations typically take the form $Ax + By = C$, where $A$, $B$, and $C$ are constants. In our exercise, the linear equation is $x + 6y = 12$. The equation depicts all possible combinations of $x$ and $y$ that satisfy the relationship defined by the equation.

Standard Form: This is the form $Ax + By = C$.
Graphing: Linear equations can be plotted on a coordinate plane as a straight line.
Application: Useful for solving problems where two quantities are compared, often finding where they intersect.

Understanding linear equations is fundamental, as they are used in various contexts like economics, physics, and everyday problem solving.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line to easily understand its slope and y-intercept. This form is expressed as $y = mx + b$, where $m$ represents the slope of the line and $b$ the y-intercept.

Slope ($m$): Indicates the steepness and direction of the line. A positive slope means the line rises, while a negative slope means it falls.
Y-intercept ($b$): Shows where the line crosses the y-axis. For example, in our exercise it's $b = 2$.
Conversion: Converting from standard form to slope-intercept form helps graph the line more efficiently.

By rewriting $x + 6y = 12$ as $y = -\frac{1}{6}x + 2$, we can more easily recognize these traits for graphing.

Inequality Graphing

Inequality graphing involves plotting a range of solutions on a graph rather than a single line. It shows all the possible solutions to an inequality, like $x + 6y \leq 12$. After removing the inequality and just graphing the line $x + 6y = 12$, you determine which side of the line satisfies the inequality by shading the appropriate region.

Boundary Line: For $\leq$ or $\geq$, the line itself is included (solid line). For $<$ or $>$, the line is dashed.
Shading: Indicate solutions either above or below the line. In this case, shade below the line because $\leq$ includes less than or equal to.

This method visually demonstrates which values of $x$ and $y$ satisfy the inequality.

Coordinate Graphing

Coordinate graphing is plotting points, lines, or regions on a plane using the coordinate system. It involves an x-axis (horizontal) and a y-axis (vertical) that intersect at the origin (0,0).

Plotting Points: Each point is defined by an $x$ and $y$ coordinate.
Drawing Lines: A line can be drawn once two points are plotted. In the exercise, the line through points based on the equation $x + 6y = 12$.
Adding Features: After plotting a line, additional features like shading can be added to represent inequalities.

Coordinate graphing is essential for visually analyzing numerical relationships, comparing linear and nonlinear relationships, and finding solutions to equations.

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91影视

Sketch the graph of the inequality. $$x+6 y \leq 12$$

Short Answer

Step by step solution

Write the inequality as an equation

Plot the line

Shade the region

Key Concepts

Linear Equations

Slope-Intercept Form

Inequality Graphing

Coordinate Graphing

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