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 (1,3).
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.
The true statements are that both boundary lines are solid, a solution to the system is (1,3), and the boundary lines intersect. Therefore, the answer is: 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 Problem We are given two inequalities: y ≤ 3 x + 1 and y ≥ − x + 2 . We need to determine which of the given statements about the graph of these inequalities are true. Let's analyze each statement.
Checking Statement 1 Statement 1: The slope of one boundary line is 2. The boundary lines are y = 3 x + 1 and y = − x + 2 . The slopes are 3 and -1 respectively. Neither slope is 2, so this statement is false.
Checking Statement 2 Statement 2: Both boundary lines are solid. Since the inequalities are ≤ and ≥ , the boundary lines are solid. This statement is true.
Checking Statement 3 Statement 3: A solution to the system is (1,3). We need to check if the point (1,3) satisfies both inequalities. Substituting x = 1 and y = 3 into the inequalities:
For y ≤ 3 x + 1 : 3 ≤ 3 ( 1 ) + 1 ⇒ 3 ≤ 4 , which is true. For y ≥ − x + 2 : 3 ≥ − ( 1 ) + 2 ⇒ 3 ≥ 1 , which is true.
Since (1,3) satisfies both inequalities, it is a solution to the system. This statement is true.
Checking Statement 4 Statement 4: Both inequalities are shaded below the boundary lines. The inequality y ≤ 3 x + 1 is shaded below the line, but the inequality y ≥ − x + 2 is shaded above the line. Therefore, this statement is false.
Checking Statement 5 Statement 5: The boundary lines intersect. To find the intersection point, we solve the system of equations:
y = 3 x + 1 y = − x + 2
Setting the two expressions for y equal to each other:
3 x + 1 = − x + 2 4 x = 1 x = 4 1
Substituting x = 4 1 into y = − x + 2 :
y = − 4 1 + 2 = 4 7
The intersection point is ( 4 1 , 4 7 ) . Since a solution exists, the lines intersect. This statement is true.
Final Answer The true statements are: Both boundary lines are solid, A solution to the system is (1,3), and The boundary lines intersect.
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, subject to constraints on resources like labor and materials. Each inequality represents a constraint, and the solution set represents the feasible production plans that satisfy all 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.
