For the linear inequality $y \leq -x+5$, you would use a solid line and shade above.
True
False
Answer (2)
The inequality is y ≤ − x + 5 .
The line should be solid because the inequality includes 'equal to'.
Testing the point (0,0) shows that the region below the line should be shaded.
Therefore, the statement 'you would use a solid line and shade above' is False. F a l se
Explanation
Analyzing the Inequality The given inequality is y ≤ − x + 5 . We need to determine whether the line should be solid or dashed and which side of the line should be shaded.
Determining the Line Type Since the inequality is y ≤ − x + 5 , which includes the 'equal to' case, the boundary line should be solid.
Determining the Shading Region To determine the shading, we can test a point. Let's test the point (0,0). Substituting into the inequality, we get 0 ≤ − 0 + 5 , which simplifies to 0 ≤ 5 . This statement is true, so the region containing (0,0) should be shaded. The region below the line should be shaded.
Comparing with the Statement The statement says that we should shade above the line. However, we found that we should shade below the line. Therefore, the statement is false.
Final Answer The statement is false.
Examples
Linear inequalities are used in various real-world applications, such as determining the feasible region in linear programming problems. For example, a company might use linear inequalities to determine the optimal production levels of two products, given constraints on resources like labor and materials. The solution to the system of inequalities would represent the set of production levels that satisfy all the constraints.
The given inequality y ≤ − x + 5 indicates a solid line should be used because it includes equal to. Testing the point (0,0) reveals we should shade below the line, not above. Thus, the statement is False.
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