Linear Inequalities

Solve $4x + 2 > 2$ for $x \in \mathbb{R}$.

Solve $5(x - 2) \leq 0$ for $x \in \mathbb{R}$.

If $x \in \mathbb{N}$, solve $x - 4 \leq -2$.

Solve $\frac{x}{10} + 1 < 1$ for $x \in \mathbb{R}$.

Find the set of all real $x$ such that $\frac{2x + 1}{3} \geq 3$.

Find the solution set of $|x - 4| < 0$.

Solve the simultaneous inequalities $x + 2 > 0$ and $2x - 5 < 0$.

Solve the inequality $\frac{x+4}{x-3} < 2$.

Which of the following describes the solution region of $x + y \le 5, x - y \ge 3, x \ge 0, y \ge 0$?

Solve for $x$: $\frac{x^2 - 1}{x^2 - 7x + 12} \le 0$.

Find the range of values of $x$ for which $x + 3 > 0$ and $2x < 14$.

Solve $22x < 100$ when $x$ is a natural number.

Solve the inequality $|x - 1| \le 2$ and identify the interval.

Solve $\frac{3(x - 2)}{5} \ge \frac{5(2 - x)}{3}$.

Which of the following represents the algebraic solution and its graphical interpretation for the inequality $\frac{x - 3}{2} + 5 \geq 4 - 2x$?

Represent the solution of $x + y \ge 5$ graphically in the two-dimensional plane. Which region does it represent relative to the line $x + y = 5$?

Solve the inequality $\frac{1}{x-2} < 0$.

A car rental company charges Rs 1000 plus Rs 15 per kilometer. If a tourist has a budget between Rs 4000 and Rs 5500 for the rental, what is the range of distance $d$ (in km) they can travel?

For a particular investment, the profit $P$ (in thousands) is modeled by $P(x) = 4x - 20$, where $x$ is the number of units sold. If the company needs to ensure a profit between 40 and 100 thousand inclusive, what is the required range for $x$?

A manufacturer of light bulbs tests a sample. The life $x$ (in hours) of the bulbs is given by the inequality $|x - 1200| < 100$. Which interval represents the range of the lifespan of these bulbs?

Which of these points lies in the solution set of $x - y \le 2$?

The inequality $y \ge -3$ consists of which points?

A student graphs $x + 2y < 4$. They test $(0,0)$ and find $0 < 4$. Where should they shade?

What is the graphical representation of $x \le 0$?

In the graphical solution of the inequality $y < -2x + 4$, the boundary line is drawn as a dashed line. Why is the boundary line dashed rather than solid?

Identify the coordinates of the vertices of the feasible region defined by the system: $x \ge 0, y \ge 0, x + 2y \le 8, 3x + 2y \le 12$.

Consider the inequality $2x - 3y > 6$. Which half-plane represents the solution set for this inequality relative to the line $2x - 3y = 6$?

A production plant makes two products using a chemical process. The reaction requires $x$ units of Chemical A and $y$ units of Chemical B. The constraints are $x + y \le 10, 2x + y \le 15, x \ge 2, y \ge 2$. What is the shape of the feasible region?

A student wants to buy books ($x$) and pens ($y$). Each book costs Rs 50 and each pen costs Rs 10. He wants to buy at least 2 books and 5 pens, but cannot spend more than Rs 300. Which point satisfies all conditions?

In the graphical solution of $x - y \le 0$ and $x + y \ge 2$, where $x, y \ge 0$, which quadrant contains the unbounded portion of the solution set?

The region $x + y < 2$ and $x + y > 3$ is:

Which of these is a solution for $2x - y \le 3$?

In the system $x \ge 1, y \ge 1$, what is the minimum value of $x+y$?

The point $(1, -5)$ satisfies which system?

If the solution to a system of inequalities is a line segment between $(1,1)$ and $(2,2)$, the system could be:

The system of inequalities $x - y \leq 2, x + y \leq 2, x \geq 0$ forms a triangle with vertices:

Which system of inequalities defines an empty set?

The minimum value of $Z = x - y$ subject to $x + y \le 5, x \ge 0, y \ge 0$ is:

The system $x + y \le 5, x - y \ge 0, y \ge 1$ defines a triangular region. Find its area.

How many integral points $(x, y)$ satisfy $x > 0, y > 0,$ and $2x + 2y \le 5$?