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Chapter 7: Problem 32

Graph the solution set of system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}y<-2 x+4 \\\y< x-4\end{array}\right.$$

Short Answer

Expert verified
The solution set for the system of inequalities will be graphically represented by the area shaded under both lines, \(y < -2x + 4\) and \(y < x - 4\).

Step by step solution

01

Analyze Each Inequality

The system has two inequalities:1. \(y < -2x + 4\)2. \(y < x - 4\)In each of these, y is less than a linear function of x. This means, for each inequality, the solution will be the region below the line formed.
02

Graphing First Inequality

The first inequality is \(y < -2x + 4\). To graph this inequality, begin by sketching the line \(y = -2x + 4\). This is a line with slope -2 and y-intercept +4. Keep in mind that since y is 'less than' the line, the line will be dashed and the shading will be below it.
03

Graphing Second Inequality

The second inequality is \(y < x - 4\). In a similar manner, graph the line \(y = x - 4\). This line has a slope of 1 and a y-intercept of -4. Remember, since y is 'less than' this line as well, the line will be dashed and shading will be below it.
04

The Solution Set

The solution for this system of inequalities is the overlapping region of solutions of these two inequalities. What is important to know is that the region of the graph that is shared by both inequalities is the solution to the system. So, check the area which is shaded by both inequalities.

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Key Concepts

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

Graphing Inequalities
Imagine inequalities as road zones. Your task is to identify which parts are `neighborhoods`. Graphing inequalities can help visualize where these neighborhoods are located. When graphing inequalities like \(y < -2x + 4\) and \(y < x - 4\), you must first understand they represent areas on a graph. Here's how to do it:
  • First, graph the line for \(y = -2x + 4\) and \(y = x - 4\). These serve as the boundaries.
  • Use a dashed line type because the inequalities are less than (<), not equal to. This shows the border can be approached but not crossed.
  • Shade the side of the line where the inequality holds true. For these equations, it's the side below the line since "y" values are less than the line equation values.
By carefully shading, you'll see the regions that represent solutions. The concept of graphing inequalities is all about marking where conditions are met on a plane.
Linear Inequality
A linear inequality is similar to a linear equation, but instead of an equal sign (=), it uses an inequality sign like < or >.
  • The inequality \(y < -2x + 4\) suggests that the value of 'y' is less than the expression \(-2x + 4\).
  • Likewise, \(y < x - 4\) indicates \'y\' is less than the expression \(x - 4\).
  • Each inequality can be seen as a rule defining a region on the graph. That's why we sketch the lines these equations form and only include regions that satisfy these `rules`.
In simpler terms, a linear inequality shows all potential values of 鈥測" that fall below a specific threshold defined by a straight-line equation. Think of it as drawing a map's borders, beyond which the inequality's condition doesn't fit.
Solution Set
The solution set of a system of inequalities is the area where all conditions and rules of the system are satisfied simultaneously. Think of it as the `common ground` where all lines agree.
  • Once both inequalities are graphed with their respective shading, find the overlapping area. This overlap is where both inequalities are true at the same time.
  • The solution set is the intersection, representing all (x, y) pairs that satisfy both inequalities.
  • If no overlapping region exists, it means the system has no solution. Fortunately, in our case of \(y < -2x + 4\) and \(y < x - 4\), such a shared region does exist.
In essence, the solution set is like a Venn diagram on a graph. It encircles only those solutions that sit in both inequality zones simultaneously. Always inspect this overlap to find the valid solution zone.

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Most popular questions from this chapter

Harsh, mandatory minimum sentences for drug offenses account for more than half the population in U.S. federal prisons. The bar graph shows the number of inmates in federal prisons, in thousands, for drug offenses and all other crimes in 1998 and \(2010 .\) (Other crimes include murder, robbery, fraud, burglary, weapons offenses, immigration offenses, racketeering, and perjury.) (GRAPH CAN'T COPY) a. In \(1998,\) there were 60 thousand inmates in federal prisons for drug offenses. For the period shown by the graph, this number increased by approximately 2.8 thousand inmates per year. Write a function that models the number of inmates, \(y,\) in thousands, for drug offenses \(x\) years after 1998. b. In \(1998,\) there were 44 thousand inmates in federal prisons for all crimes other than drug offenses. For the period shown by the graph, this number increased by approximately 3.8 thousand inmates per year. Write a function that models the number of inmates, \(y,\) in thousands, for all crimes other than drug offenses \(x\) years after 1998 . c. Use the models from parts (a) and (b) to determine in which year the number of federal inmates for drug offenses was the same as the number of federal inmates for all other crimes. How many inmates were there for drug offenses and for all other crimes in that year?

Sketch the graph of the solution set for the following system of inequalities: $$\left\\{\begin{array}{l}y \geq n x+b(n<0, b>0) \\\y \leq m x+b(m>0, b>0)\end{array}\right.$$

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