Problem 9 - Elementary Algebra Within Reach

Question

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} y

Step-by-Step Solution

Answer

The graph of the system of inequalities consists of the area that is to the left of the vertical line \(x=5\) and below the sloped line \(y=x-2\).

Step 1: Graph y = x - 2

This equation is a straight line with a slope of 1 and y-intercept at -2. Draw this line on graph paper. Since equation is \(y

Step 2: Graph x = 5

This is a vertical line passing through the x-axis at 5. Draw this line on the same graph. Since equation is \(x<5\), shade the area to the left of the line.

Step 3: Find the intersection of the two shaded regions

The intersection of these two regions represents the set of points that satisfy both inequalities. Thus, the shaded area to the left of the vertical line and below the straight line represents the solution to this system of inequalities.

Key Concepts

Understanding Systems of Inequalities

When we talk about systems of inequalities, we are referring to a set of two or more inequalities that we consider together. Solving these systems involves finding all the solutions that satisfy each inequality in the system. Think of it as overlapping conditions that a solution must meet. In our given exercise, we have two inequalities:

• \(y < x - 2\)
• \(x < 5\)

Each of these inequalities describes a half-plane on the graph. The solution to the system will be the overlapping region where the solutions to all these inequalities come together.

Remember, the overlapping area where both conditions are satisfied is the solution to the system of inequalities. Understanding this concept is crucial, as it provides a visual and analytical way to grasp the solution set of such systems right at once.

Graphing Linear Equations for Systems

To effectively work with systems of inequalities, understanding how to graph linear equations is necessary. A linear equation like \(y = x - 2\) can be rewritten in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

For example, in the equation \(y = x - 2\):

• The slope \(m\) is 1, which means for every step increase in \(x\), \(y\) increases by 1.
• The y-intercept \(b\) is -2, where the line crosses the y-axis.

To graph this:1. Start from the y-intercept (-2 on the y-axis).2. Use the slope to determine the direction of the line. In this case, move up 1 unit for every rightward movement of 1 unit along the x-axis.3. Draw a dashed line since the inequality is \(y < x - 2\) (less than) and not \(\leq\).
Next, tackle \(x = 5\):

• This is a vertical line crossing the x-axis at 5.
• Graph it as a straight line parallel to the y-axis and draw a dashed line for \(x < 5\).

Proper graphing allows you to see the boundaries and decide which regions to shade.

Finding Inequality Solutions

The solution to a system of linear inequalities is represented by the region where the shaded areas overlap. This region contains all the points that satisfy each and every inequality in the system.

In practical terms, follow these steps:

• Graph each inequality on the same coordinate plane.
• For the inequality \(y < x - 2\), shade the region below the line because \(y\) is less than \(x - 2\).
• For the inequality \(x < 5\), shade the area to the left because \(x\) should be less than 5.
• The solution set is the intersection of these two shaded regions.

Remember, use dashed lines for inequalities without an equal sign. The boundary lines themselves are not part of the solution. Only the overlapping shaded area counts. This helps visually and methodically confirm the set of solutions to the system.

Previous

Problem 8

Next

Problem 9

Other exercises in this chapter

Problem 8

In Exercises 5-14, solve the system by the method of substitution. $$ \left\\{\begin{array}{r} 2 x-y=0 \\ 3 y=6 \end{array}\right. $$

View solution

Problem 8

In Exercises \(5-10\), solve the system by graphing. $$ \left\\{\begin{array}{l} y=\frac{1}{2} x+2 \\ y=-x+8 \end{array}\right. $$

View solution

Problem 9

In Exercises \(7-10\), use a system of linear equations to solve the problem. The sale price of a microwave oven is \(\$ 110\). The discount is \(20 \%\) of the

View solution

Problem 9

In Exercises 7-12, solve the system by the method of elimination. $$ \left\\{\begin{array}{r} -x+2 y=6 \\ 2 x+5 y=6 \end{array}\right. $$

View solution

Key Concepts

Understanding Systems of Inequalities

When we talk about systems of inequalities, we are referring to a set of two or more inequalities that we consider together. Solving these systems involves finding all the solutions that satisfy each inequality in the system. Think of it as overlapping conditions that a solution must meet. In our given exercise, we have two inequalities:

• \(y < x - 2\)
• \(x < 5\)

Each of these inequalities describes a half-plane on the graph. The solution to the system will be the overlapping region where the solutions to all these inequalities come together.

Remember, the overlapping area where both conditions are satisfied is the solution to the system of inequalities. Understanding this concept is crucial, as it provides a visual and analytical way to grasp the solution set of such systems right at once.

Graphing Linear Equations for Systems

To effectively work with systems of inequalities, understanding how to graph linear equations is necessary. A linear equation like \(y = x - 2\) can be rewritten in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

For example, in the equation \(y = x - 2\):

• The slope \(m\) is 1, which means for every step increase in \(x\), \(y\) increases by 1.
• The y-intercept \(b\) is -2, where the line crosses the y-axis.

To graph this:1. Start from the y-intercept (-2 on the y-axis).2. Use the slope to determine the direction of the line. In this case, move up 1 unit for every rightward movement of 1 unit along the x-axis.3. Draw a dashed line since the inequality is \(y < x - 2\) (less than) and not \(\leq\).
Next, tackle \(x = 5\):

• This is a vertical line crossing the x-axis at 5.
• Graph it as a straight line parallel to the y-axis and draw a dashed line for \(x < 5\).

Proper graphing allows you to see the boundaries and decide which regions to shade.

Finding Inequality Solutions

The solution to a system of linear inequalities is represented by the region where the shaded areas overlap. This region contains all the points that satisfy each and every inequality in the system.

In practical terms, follow these steps:

• Graph each inequality on the same coordinate plane.
• For the inequality \(y < x - 2\), shade the region below the line because \(y\) is less than \(x - 2\).
• For the inequality \(x < 5\), shade the area to the left because \(x\) should be less than 5.
• The solution set is the intersection of these two shaded regions.

Remember, use dashed lines for inequalities without an equal sign. The boundary lines themselves are not part of the solution. Only the overlapping shaded area counts. This helps visually and methodically confirm the set of solutions to the system.