Which statements are true about the graph of [tex]$y \leq 3 x+1$[/tex] and [tex]$y \geq-x+2$[/tex]? Check all that apply.
A. The slope of one boundary line is 2.
B. Both boundary lines are solid.
C. A solution to the system is [tex]$(1,3)$[/tex].
D. Both inequalities are shaded below the boundary lines.
E. The boundary lines intersect.
Answer (1)
The slope of one boundary line is 2: False.
Both boundary lines are solid: True.
A solution to the system is ( 1 , 3 ) : True.
Both inequalities are shaded below the boundary lines: False.
The boundary lines intersect: True.
Therefore, the true statements are that both boundary lines are solid, a solution to the system is (1,3), and the boundary lines intersect. B o t hb o u n d a ry l in es a reso l i d , A so l u t i o n t o t h esys t e mi s ( 1 , 3 ) , T h e b o u n d a ry l in es in t ersec t
Explanation
Analyzing the Inequalities We are given two inequalities: y ≤ 3 x + 1 and y ≥ − x + 2 . We need to determine which statements about their graphs are true. Let's analyze each statement.
Checking the Slopes The first statement says 'The slope of one boundary line is 2'. The boundary lines are y = 3 x + 1 and y = − x + 2 . The slopes of these lines are 3 and -1, respectively. Since neither slope is 2, this statement is false.
Checking Boundary Lines The second statement says 'Both boundary lines are solid'. Since the inequalities are y ≤ 3 x + 1 and y ≥ − x + 2 , both include the '=' sign. This means the boundary lines are solid. So, this statement is true.
Checking the Solution The third statement says 'A solution to the system is (1,3)'. To check this, we substitute x = 1 and y = 3 into both inequalities: For y ≤ 3 x + 1 , we have 3 ≤ 3 ( 1 ) + 1 , which simplifies to 3 ≤ 4 . This is true. For y ≥ − x + 2 , we have 3 ≥ − ( 1 ) + 2 , which simplifies to 3 ≥ 1 . This is true. Therefore, (1,3) is a solution to the system, and this statement is true.
Checking the Shading The fourth statement says 'Both inequalities are shaded below the boundary lines'. The inequality y ≤ 3 x + 1 is shaded below the boundary line because it includes all points where the y-value is less than or equal to 3 x + 1 . However, the inequality y ≥ − x + 2 is shaded above the boundary line because it includes all points where the y-value is greater than or equal to − x + 2 . Therefore, this statement is false.
Checking Intersection The fifth statement says 'The boundary lines intersect'. To check this, we can solve the system of equations: y = 3 x + 1 y = − x + 2 Setting the two expressions for y equal to each other, we get: 3 x + 1 = − x + 2 4 x = 1 x = 4 1 Substituting this value of x into either equation, we can find the value of y: y = 3 ( 4 1 ) + 1 = 4 3 + 1 = 4 7 Since we found a solution for x and y, the boundary lines intersect. Therefore, this statement is true.
Examples
Understanding systems of inequalities is crucial in various real-world applications. For instance, consider a scenario where a company needs to optimize its production of two products, say tables and chairs. Each product requires a certain amount of resources like wood and labor. The inequalities can represent the constraints on the available resources, such as the maximum amount of wood or labor hours. The solution set of the system of inequalities would then represent the feasible production plans that satisfy all resource constraints. By graphing these inequalities, the company can visually identify the region of feasible solutions and make informed decisions about production levels to maximize profit while staying within the resource limits. This approach is fundamental in operations research and management science.
