Linear inequalities in 1 or 2 variables：练习题

What is the solution to the inequality \( 4x - 7 \leq 13 \)?

In the \( xy \)-plane, the solution set to a system of inequalities is the region where the graphs of \( y \leq -x + 5 \) and \( y \geq 2x - 4 \) overlap. Which of the following points \( (x, y) \) lies in this solution set?

A graph in the \( xy \)-plane shows the feasible region for a business constraint. The business produces \( x \) units of Item A and \( y \) units of Item B. Each Item A requires 2 labor hours and each Item B requires 3 labor hours. The total labor available is at most 60 hours. The business must also produce at least 15 total units. Which system of inequalities represents these constraints?

What is the set of all solutions to the inequality \( 3(x - 4) < 5x + 2 \)?

Which of the following points \( (x, y) \) is a solution to the system of inequalities \( y > 2x - 1 \) and \( y < -x + 4 \)?

What is the smallest integer value of \( x \) that satisfies the inequality \( \frac{2x - 5}{3} \ge \frac{x + 7}{2} \)?

A region in the \( xy \)-plane is defined by the system of inequalities \( y \ge 0 \), \( x \ge 0 \), and \( x + 2y \le 8 \). What is the maximum possible value of the expression \( 5x + 4y \) for any point \( (x, y) \) within this region?

A system of linear inequalities is defined by:
\( y > x - 5 \)
\( y < -x + 5 \)

Part a: Determine if the point \( (4, 2) \) is a solution to the system. Provide calculations for both inequalities to justify your answer.
Part b: Find the \( x \)-coordinate of the point where the boundary lines \( y = x - 5 \) and \( y = -x + 5 \) intersect.
Part c: For a point in the solution set with an \( x \)-coordinate of 2, determine the range of all possible \( y \)-values.

A triangular region in the \(xy\)-plane is defined by the following system of linear inequalities:
1) \(y \le 2x + 4\)
2) \(y \ge -x + 1\)
3) \(x \le 3\)

a) Find the coordinates of the three vertices that form the boundary of this triangular region.
b) Calculate the area of the triangular region formed by these inequalities.
c) Determine the maximum value of the expression \(P = 3x + y\) for any point \((x, y)\) within this shaded region.<\/p>