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.

The true statements regarding the graph of the inequalities are: both boundary lines are solid, the point (1,3) is a solution, and the boundary lines intersect. The slopes are not 2, and both inequalities are not shaded below the boundaries.
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Answered by Anonymous | 2025-07-04

The slope of y = 3 x + 1 is 3, so the first statement is false.
Both inequalities include the '=' sign, so both boundary lines are solid.
The point (1,3) satisfies both inequalities, so it is a solution.
y ≤ 3 x + 1 is shaded below, and y g e q − x + 2 is shaded above, so the fourth statement is false.
Solving 3 x + 1 = − x + 2 gives x = 0.25 and y = 1.75 , so the lines intersect. Thus, the final answer is: Both boundary lines are solid, A solution to the system is (1,3), The boundary lines intersect. ​

Explanation

Analyzing the Inequalities We are given two inequalities: y ≤ 3 x + 1 and y g e q − x + 2 . We need to determine which statements about the graph of these inequalities are true. Let's analyze each statement.

Checking the Slopes The first boundary line is y = 3 x + 1 . The slope of this line is 3, not 2. So, the statement 'The slope of one boundary line is 2' is false. The second boundary line is y = − x + 2 . The slope of this line is -1.

Checking Boundary Lines Since the inequalities are y ≤ 3 x + 1 and y g e q − x + 2 , both inequalities include the '=' sign. This means both boundary lines are solid. So, the statement 'Both boundary lines are solid' is true.

Checking the Solution Let's check if the point (1,3) is a solution to the system. For the first inequality, 3 ≤ 3 ( 1 ) + 1 = 4 , which is true. For the second inequality, 3 g e q − ( 1 ) + 2 = 1 , which is also true. So, the statement 'A solution to the system is (1,3)' is true.

Checking the Shading The inequality y ≤ 3 x + 1 is shaded below the boundary line because the y-values are less than or equal to the values on the line. The inequality y g e q − x + 2 is shaded above the boundary line because the y-values are greater than or equal to the values on the line. So, the statement 'Both inequalities are shaded below the boundary lines' is false.

Checking Intersection To find the intersection point of the boundary lines, we set 3 x + 1 = − x + 2 . Solving for x, we get 4 x = 1 , so x = f r a c 1 4 = 0.25 . Then, y = 3 ( 0.25 ) + 1 = 0.75 + 1 = 1.75 . Also, y = − 0.25 + 2 = 1.75 . Since the x and y values exist, the boundary lines intersect. So, the statement 'The boundary lines intersect' is true.

Final Answer Therefore, the true statements are:



Both boundary lines are solid.
A solution to the system is (1,3).
The boundary lines intersect.

Examples
Systems of inequalities are used in various real-world applications, such as linear programming, where you might want to optimize a certain objective function subject to constraints. For example, a company might want to maximize its profit given constraints on resources like labor and materials. The inequalities represent these constraints, and the solution to the system represents the feasible region where all constraints are satisfied. By graphing these inequalities, the company can visually identify the region of possible solutions and determine the optimal production strategy.

Answered by GinnyAnswer | 2025-07-04