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Chapter 8: Problem 56

Graph the solution set of each system of inequalities. $$\left\\{\begin{array}{l} x \quad \leq 10 \\ x+y \geq 7 \end{array}\right.$$

Short Answer

Expert verified
The solution set is the region above the line of the second inequality but to the left side of the line of the first inequality.

Step by step solution

01

Graph the first inequality

The first inequality states that \( x \) is less than or equal to 10. It represents a vertical line running through \( x = 10 \) and includes all points to the left of this line.
02

Graph the second inequality

The second inequality is \( x+y \geq 7 \). This is slightly more complex as it involves two variables. First, arrange it into slope-intercept form, i.e., \( y \geq -x + 7 \). This will be a line that crosses the y-axis at 7 and has a negative slope. The inequality sign indicates a greater or equal condition, thus shading should be performed above the line.
03

Combine the inequalities

Having graphed both inequalities, the shared region (area common to both shaded areas) represents the solution set to the system of inequalities. This will cover the area above the second line (the line from Step 2) but restricted to the left of the first line (the vertical line from Step 1).

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Key Concepts

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

Solution Set
When working with systems of inequalities, finding the solution set is a key step. A solution set consists of all the points that satisfy each inequality in the system.
For our example, the solution set will be the overlapping region on the coordinate plane where both conditions are true.
This involves graphing each inequality separately and identifying where the shaded regions intersect.
  • For the inequality \( x \leq 10 \), we consider all points to the left of the vertical line at \( x = 10 \).
  • For the inequality \( x+y \geq 7 \), we shade the region above the line \( y = -x + 7 \).
We are looking for the portion of the graph that is shaded by both inequalities.
This shared shading represents the solution set. Visualizing this will help to confirm that all points in this region satisfy both constraints.
System of Inequalities
A system of inequalities involves two or more inequalities that are considered at the same time. Each inequality restricts a different combination of variables.
In our exercise, we have a system of two inequalities in two variables:
  • The first inequality \( x \leq 10 \) focuses on an upper boundary for the \( x \) values.
  • The second inequality \( x+y \geq 7 \) establishes a minimum bound created through combining \( x \) and \( y \).
The goal of solving a system of inequalities is to find all possible solutions that satisfy both conditions simultaneously.
To solve graphically, plot each inequality on a coordinate plane. The overlapping shaded area from each inequality forms the solution to the system
This highlights where both conditions are met for values of \( x \) and \( y \).
Slope-Intercept Form
The slope-intercept form of a line is one of the most common ways to express linear equations.
In this format, a line can be written as:\[ y = mx + b \]where:
  • \( m \) represents the slope of the line, indicating its steepness.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
The inequality \( x+y \geq 7 \) can be rearranged into slope-intercept form as \( y \geq -x + 7 \).
In this representation:
  • The slope \( m = -1 \) indicates the downward direction of the line.
  • The intercept \( b = 7 \) shows where the line intersects the y-axis.
Graphing this line with shading above it determines the region that satisfies the inequality.
Understanding the slope-intercept form is crucial for easily converting inequalities, and helps in determining the shading direction needed for inequalities as well.

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Most popular questions from this chapter

The area of a rectangular property is 1800 square feet; its length is twice its width. There is a rectangular swimming pool centered within the property. The dimensions of the property are one and onethird times the corresponding dimensions of the pool. The portion of the property that lies outside the pool is paved with concrete. What are the dimensions of the property and of the pool? What is the area of the paved portion?

Decode the message, which was encoded using the matrix \(\left[\begin{array}{rrr}1 & -2 & 3 \\ -2 & 3 & -4 \\ 2 & -4 & 5\end{array}\right]\). $$\left[\begin{array}{r}-5 \\\0 \\\\-11\end{array}\right],\left[\begin{array}{r}20 \\\\-36 \\\38\end{array}\right]$$

Find the decoding matrix for each encoding matrix. $$\left[\begin{array}{rrrr}1 & 0 & -1 & 1 \\\\-2 & 3 & 3 & 7 \\\2 & 0 & -6 & 0 \\\\-1 & 1 & 2 & 2\end{array}\right]$$

Joi and Cheyenne are planning a party for at least 50 people. They are going to serve hot dogs and hamburgers. Each hamburger costs \(\$ 1\) and each hot dog costs \(\$ .50 .\) Joi thinks that each person will eat only one item, either a hot dog or a hamburger. She also estimates that they will need at least 15 hot dogs and at least 20 hamburgers. How many hamburgers and how many hot dogs should Joi and Cheyenne buy if they want to minimize their cost?

Involve the use of matrix multiplication to transform one or more points. This technique, which can be applied to any set of points, is used extensively in computer graphics. Consider a series of points \(\left(x_{0}, y_{0}\right),\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) such that, for every nonnegative integer \(i,\) the point \(\left(x_{i+1}, y_{i+1}\right)\) is found by applying the matrix \(\left[\begin{array}{cc}1 & -2 \\ 1 & -3\end{array}\right]\) tothe point \(\left(x_{i}, y_{i}\right)\) $$\left[\begin{array}{l}x_{i+1} \\\y_{i+1}\end{array}\right]=\left[\begin{array}{ll}1 & -2 \\\1&-3\end{array}\right]\left[\begin{array}{l}x_{i} \\\y_{i}\end{array}\right]$$ (a) Find \(\left(x_{1}, y_{1}\right)\) if \(\left(x_{0}, y_{0}\right)=(2,-1)\) (b) Find \(\left(x_{2}, y_{2}\right)\) if \(\left(x_{0}, y_{0}\right)=(4,6) .\) (Hint: Find \(\left(x_{1}, y_{1}\right)\) first.) (c) Use the inverse of an appropriate matrix to find \(\left(x_{0}, y_{0}\right)\) if \(\left(x_{3}, y_{3}\right)=(2,3)\)
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