Practice Questions

Identify which of the following is a 'double inequality': a) 3x + 5 < 10, b) 2 <= x < 9, c) y > 4x - 1.

Formulate a double inequality to represent the following statement: 'For a scientific experiment to be valid, the atmospheric pressure $P$ (in kPa) must be greater than 100 kPa but no more than 105 kPa.'

Define a 'numerical inequality' and provide an example.

Design a word problem involving a student's test scores that results in the inequality $\frac{85 + 91 + x}{3} \geq 90$. Solve for $x$ and interpret the result in the context of your problem.

List the four main inequality symbols used to form an inequality.

Define the term 'linear inequality' in the context of algebra.

Explain how to find the solution to a system of two linear inequalities in one variable, such as 'x > 2' and 'x <= 5'.

Explain the difference between a 'strict inequality' and a 'slack inequality', providing one example for each.

List the four general forms of a linear inequality in one variable, x, where a is not equal to zero.

Recall the effect on the inequality sign when both sides of the inequality 'x > y' are divided by -1.

Evaluate the statement: 'The solution set for the inequality $x^2 + 4 < 0$ is the empty set, $\phi$'. Justify your evaluation for all real numbers $x$.

A student solved the inequality $-3(x-2) \geq 9$ and their first step was to write $x-2 \geq -3$. Critique the student's reasoning in this step and explain the correct procedure.

Justify why the inequality sign must be reversed when multiplying both sides by a negative number. Use a numerical example other than those presented in the source text.

Describe the two fundamental rules for solving a linear inequality.

Explain what is meant by the 'solution' of an inequality in one variable. Provide the solution set for 'x < 5' where x is a natural number.

Name the set of values of the variable that make an inequality a true statement.

Summarize the key difference in the procedure for solving an inequality compared to solving a linear equation, especially concerning multiplication and division. Explain why this difference is necessary using a numerical example.

Propose a real-world scenario, different from purchasing items, that can be modeled by the linear inequality $250x + 400y \leq 3000$, where $x$ and $y$ must be non-negative integers.

Describe why a linear inequality in one variable, when solved over the set of real numbers, often has an infinite number of solutions.

Summarize the basic steps to solve a word problem that translates into a linear inequality, such as finding the minimum score needed on a test to achieve an average.

Describe the complete process of representing the solution of a linear inequality in one variable on a number line. Explain the graphical difference between representing 'x < 3' and 'x ≤ 3'.