Practice Questions

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

Define a 'numerical inequality' and provide an example.

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.'

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.

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

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

Describe the two fundamental rules for solving a linear inequality.

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 $x$.

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.

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

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.

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.

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

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.

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.

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.

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.

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'.

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