Question

The cost of each table or each chair cannot exceed \(\mathrm{Rs} 9\). If the cost of 4 tables and 5 chairs cannot exceed Rs 120 , then the inequations which best represents the above information are (1) \(x<9, y<9,5 x+4 x \geq 120\) (2) \(x>9, y>9,4 x+5 y \geq 120\) (3) \(0

Short answer

Answer: (3) \(0<x \leq 9,0<y \leq 9,4 x+5 y \leq 120\)

Step-by-step solution

Writing Inequalities for the Problem

We will now write down the inequalities corresponding to the constraints mentioned above: 1. Cost of table (x) cannot exceed Rs 9: \(0 < x \leq 9\) 2. Cost of chair (y) cannot exceed Rs 9: \(0 < y \leq 9\) 3. Cost of 4 tables and 5 chairs cannot exceed Rs 120: \(4x + 5y \leq 120\) Now let's check which option matches with our inequalities.

Verifying the Options

(1) \(x<9, y<9,5 x+4 x \geq 120\): This option has incorrect inequalities (wrong direction). (2) \(x>9, y>9,4 x+5 y \geq 120\): This option directly contradicts the given constraints regarding the maximum price for the table (x) and chair (y), given they must not exceed Rs 9. (3) \(0<x \leq 9,0<y \leq 9,4 x+5 y \leq 120\): This option matches our derived inequalities and correctly represents the given constraints in the problem. So this is the correct option. (4) \(0<x \leq 9,0<y \leq 9,5 x+4 y \geq 120\): This option has an incorrect third inequality (wrong direction). Therefore, the correct inequality system is option (3) \(0<x \leq 9,0<y \leq 9,4 x+5 y \leq 120\).

Key concepts

inequality constraints

Inequality constraints are fundamental in problems requiring limitations or restrictions within certain boundaries. In the context of this exercise, the term refers to the conditions that must be satisfied by the cost of tables and chairs. We have two primary constraints:

• The cost of each table, denoted as \(x\), must be more than Rs 0 but not exceed Rs 9. This is mathematically expressed as \(0 < x \leq 9\).
• Similarly, the cost of each chair, denoted as \(y\), also needs to fit within the same boundary: \(0 < y \leq 9\).

These inequality constraints provide us with the range of acceptable values for \(x\) and \(y\) and ensure that the costs do not surpass the specified limits. By defining these inequalities, we can focus on values that meet these requirements, discarding any irrelevant scenarios that don't fulfill these criteria. Thus, inequality constraints help in effectively narrowing down viable options.

mathematical reasoning

Mathematical reasoning is the process of using logical thinking to analyze and solve problems. It requires making connections between given information and the conclusions to be drawn. In this exercise, we use mathematical reasoning to determine which inequalities properly represent the conditions given.
First, we identify and write down all relevant inequations based on the prompt. We know the allowable cost ranges for tables and chairs, and we also know an additional combined cost limitation:

• Both \(x\) and \(y\) need to fit the constraint \(0 < x \leq 9\) and \(0 < y \leq 9\).
• The total cost of 4 tables and 5 chairs cannot exceed Rs 120, which is represented by \(4x + 5y \leq 120\).

We then compare the given multiple-choice inequalities against the derived conditions. The option that fulfills all conditions is the correct answer. Here, the correct choice is \(0 < x \leq 9, 0 < y \leq 9, 4x + 5y \leq 120\), as it adheres to the constraints imposed while respecting the maximum accumulated budget for the items.

systems of inequalities

Systems of inequalities consist of multiple inequalities that have to be satisfied simultaneously. These systems can model complex situations with various constraints. By solving these systems, we find possible solutions that fit all the inequalities simultaneously.
In this exercise, we are dealing with a system consisting of three inequalities:

• The individual cost of each table \(0 < x \leq 9\).
• The individual cost of each chair \(0 < y \leq 9\).
• The total budget constraint for 4 tables and 5 chairs, \(4x + 5y \leq 120\).

To solve such systems, we need to identify a set of \(x\) and \(y\) values that satisfy all the inequalities when considered together. Typically, graphing these inequalities on a coordinate plane and looking at the intersecting region can help visualize feasible solutions. The region where all inequalities overlap is the solution set satisfying all conditions within the system. In practical terms, this systematic approach helps ensure that all restrictions are considered and feasible options are clearly identified.