Problem 34
Question
Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} x+y \leq 4 \\ y \geq 2 x-4 \end{array}\right. $$
Step-by-Step Solution
Verified Answer
The system of inequalities \(x + y \leq 4\) and \(y \geq 2x - 4\) has a feasible region as a solution, which can be graphically represented by shading the overlapping region of the individual inequalities
1Step 1: Plot the first inequality
The first inequality is \(x + y \leq 4\). This can be written as \(y \leq 4 - x\) in slope-intercept form (y = mx + c). This inequality represents a downward-sloping line with a y-intercept of 4 and x-intercept of 4.
2Step 2: Plot the second inequality
The second inequality is \(y \geq 2x - 4\). This inequality represents an upward-sloping line with a y-intercept of -4 and x-intercept of 2.
3Step 3: Identify the intersection and feasible region
The intersection of these two inequalities will form a feasible region. This region is the set of points that satisfy both inequalities. To find the intersection, set the two inequalities equal to each other and solve for x and Y. This intersection point will be one of the vertices of the feasible region.
4Step 4: Identify the region that satisfies both inequalities
The feasible region that satisfies both inequalities is the intersection of the regions determined by each individual inequality. For the first inequality \(y \leq 4 - x\), the feasible region is below the line, and for the second inequality \(y \geq 2x - 4\), the feasible region is above the line. The overlapping region which satisfies both inequalities is the solution.
Key Concepts
Graphing InequalitiesFeasible RegionSlope-Intercept FormIntersection of Inequalities
Graphing Inequalities
Graphing inequalities involves plotting a region on the coordinate plane that satisfies an inequality. Inequalities describe ranges of possible solutions, as opposed to equations which pinpoint exact solutions.
To graph an inequality, first convert it to the slope-intercept form: \( y = mx + c \). This makes it easy to visualize how the line behaves.
Next, draw the line representing the equation. If the inequality is "less than or equal to" (\( \leq \)) or "greater than or equal to" (\( \geq \)), draw a solid line, indicating that points on the line are included in the solution. For "less than" (\( < \)) or "greater than" (\( > \)), use a dashed line, showing that the line itself is not part of the solution.
After plotting the line, determine which side of the line represents the solution by testing a point not on the line. If it satisfies the inequality, shade that side to indicate the solution region.
To graph an inequality, first convert it to the slope-intercept form: \( y = mx + c \). This makes it easy to visualize how the line behaves.
Next, draw the line representing the equation. If the inequality is "less than or equal to" (\( \leq \)) or "greater than or equal to" (\( \geq \)), draw a solid line, indicating that points on the line are included in the solution. For "less than" (\( < \)) or "greater than" (\( > \)), use a dashed line, showing that the line itself is not part of the solution.
After plotting the line, determine which side of the line represents the solution by testing a point not on the line. If it satisfies the inequality, shade that side to indicate the solution region.
Feasible Region
The feasible region is the area on a graph where all conditions (inequalities) are met simultaneously.
It represents all possible solutions to a system of inequalities.
To find the feasible region, first graph each inequality carefully. The area where all shaded regions overlap is the feasible region.
This region can be finite (bounded) or infinite (unbounded). In bounded feasible regions, the solution is a polygon-shaped area where the sides are lines from inequalities.
However, if regions do not overlap at all, there is no feasible region. This means that there is no solution that satisfies all inequalities in the system.
Checking the feasible region helps understand possible solutions and their limits within constraints provided by the inequalities.
To find the feasible region, first graph each inequality carefully. The area where all shaded regions overlap is the feasible region.
This region can be finite (bounded) or infinite (unbounded). In bounded feasible regions, the solution is a polygon-shaped area where the sides are lines from inequalities.
However, if regions do not overlap at all, there is no feasible region. This means that there is no solution that satisfies all inequalities in the system.
Checking the feasible region helps understand possible solutions and their limits within constraints provided by the inequalities.
Slope-Intercept Form
Slope-intercept form, written as \( y = mx + c \), is a way of expressing linear equations.
It is called a slope-intercept form because it directly reveals two critical properties of the line:
By identifying the slope, one knows whether the line rises or falls as one moves from left to right.Meanwhile, the y-intercept shows where the line begins on the y-axis.
For example, in the inequality \( y \leq 4 - x \), the slope \( m \) is \(-1\) and y-intercept \( c \) is \(4\). Hence, the line slopes downwards and starts at 4 on the y-axis.
It is called a slope-intercept form because it directly reveals two critical properties of the line:
- "m" represents the slope of the line, which is the rate at which y increases as x increases.
- "c" represents the y-intercept, the point where the line crosses the y-axis.
By identifying the slope, one knows whether the line rises or falls as one moves from left to right.Meanwhile, the y-intercept shows where the line begins on the y-axis.
For example, in the inequality \( y \leq 4 - x \), the slope \( m \) is \(-1\) and y-intercept \( c \) is \(4\). Hence, the line slopes downwards and starts at 4 on the y-axis.
Intersection of Inequalities
The intersection of inequalities is the point or region that meets the criteria of multiple inequalities at once.
To find the intersection, first solve each inequality and graph them on the same coordinate plane. Look for regions where the solutions to all inequalities overlap.
The intersection can sometimes be a single point where two lines meet, or it can be a larger region forming part of the feasible region.
To find the exact intersection point of two lines, set the equations equal to each other and solve for the variables. This involves solving the system of linear equations to uncover the exact coordinates.
These coordinates are often crucial in identifying vertices of the feasible region or confirming the presence of a solution.
To find the intersection, first solve each inequality and graph them on the same coordinate plane. Look for regions where the solutions to all inequalities overlap.
The intersection can sometimes be a single point where two lines meet, or it can be a larger region forming part of the feasible region.
To find the exact intersection point of two lines, set the equations equal to each other and solve for the variables. This involves solving the system of linear equations to uncover the exact coordinates.
These coordinates are often crucial in identifying vertices of the feasible region or confirming the presence of a solution.
Other exercises in this chapter
Problem 34
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