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Chapter 7: Problem 34

Graph each inequality. SEE EXAMPLE 2. (OBJECTIVE 2) $$y \geq 3-x$$

Short Answer

Expert verified
The graph of the inequality \(y \geq 3 - x\) is a solid line passing through (0,3) with a slope of -1, and the area above this line is shaded to indicate the solution to the inequality.

Step by step solution

01

Expressing inequality in slope-intercept form

The inequality is already in the slope-intercept form \((y = mx + b)\), where the slope \((m)\) is -1 and the y-intercept \((b)\) is 3
02

Graphing the equation part of the inequality

Start by plotting the y-intercept at (0,3) on the graph. Since the slope is -1, move 1 unit down (since its negative) and 1 unit to the right to plot the next point. Draw a line through these points. Because it is a 'greater than or equal to' inequality, the line will be solid, indicating that points on it are included as solutions.
03

Shading the appropriate area

The inequality is \(y \geq 3 - x\), which means the area to be shaded is above the line. Therefore, shade the area above the line.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a way to write the equation of a line so that its characteristics are clear and easy to understand. It is expressed as \( y = mx + b \), where:
  • \( m \) represents the slope of the line. It tells us how steep the line is, and the direction it is going (up or down).
  • \( b \) represents the y-intercept. This is the point where the line crosses the y-axis.
In the inequality \( y \geq 3 - x \), we can see that it is already in the slope-intercept form, where the slope \( m \) is -1, and the y-intercept \( b \) is 3. This means that for each step to the right, the line goes down by one unit.
Linear Inequalities
Linear inequalities are similar to linear equations but involve inequality symbols such as \( >, <, \geq, \leq \) instead of an equal sign. These symbols indicate a range of values rather than a precise value.
  • \( y \geq 3 - x \) tells us that the values of \( y \) are not just along a line but are greater than or equal to the expression on the right-hand side.
  • If the inequality signs were just \( > \) or \( < \), the line on the graph would be dashed to illustrate that the points directly on the line aren't included in the solution.
In this case, the \( \geq \) sign means the line itself is part of the solution, so it will be drawn solidly on the graph.
Plotting Points
Plotting points is an essential step in graphing any equation or inequality. This process involves using the slope and y-intercept to identify points on the graph that lie on the line that represents the inequality.
Start with the y-intercept:
  • Locate the point (0,3) on the graph. This is where the line crosses the y-axis.
Next, use the slope to find more points:
  • Since the slope is -1, move down 1 unit and 1 unit to the right to find another point. This will take you to (1,2).
  • Draw a straight line through these points. This line represents the boundary of the solution region.
Shading Regions
Shading regions is a crucial step when graphing inequalities as it shows all the possible solutions. The area to be shaded depends on the sign of the inequality.For \( y \geq 3 - x \), the solution includes all points above or on the line. This is because \( y \) values are greater or equal to those on the line:
  • If you are dealing with \( \geq \) or \( > \), shade above the line.
  • For \( \leq \) or \( < \), shade below the line.
In this scenario, the solid line, along with the shaded area above it, indicates that all these points satisfy the inequality, meaning they are part of the solution set.

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Most popular questions from this chapter

Graph the solution. $$3 x+2 y \geq 12$$

Graph the solution set of each system of inequalities, when possible. SEE EXAMPLE 7. (OBJECTIVE 4) $$\left\\{\begin{array}{l} x<-2 \\ y \geq 3 \end{array}\right.$$

Use elimination to solve each system. $$\left\\{\begin{array}{l}3(x-2)=4 y \\\2(2 y+3)=3 x\end{array}\right.$$

Graph the solution set of each system of inequalities, when possible. If not possible, state \(0 .\) SEE EXAMPLE 8. (OBJECTIVE 4) $$\begin{aligned}&y \leq-\frac{4}{3} x-2\\\&4 x+3 y>15\end{aligned}$$

Explain when a system of inequalities will have no solutions.
See all solutions

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