Q: What is interval notation?

Interval notation is a way to represent solution sets using brackets. For x > 2: (2, ∞). For x ≥ 2: [2, ∞). Parentheses ( ) mean 'not including' (open), brackets [ ] mean 'including' (closed). Infinity always uses parentheses: (-∞, 2) or (2, ∞).

Q: Can linear inequalities have no solution or infinite solutions?

Yes. Some inequalities have no solution (like x > 5 and x < 3 simultaneously - impossible). Others have all real numbers as solutions (like x + 1 > x, which simplifies to 1 > 0, always true). Most have a solution set that's a ray or half-line.

Question 1

What is a linear inequality?

Accepted Answer

A linear inequality is a mathematical statement comparing two expressions using inequality symbols: > (greater than), < (less than), ≥ (greater than or equal), or ≤ (less than or equal). Linear inequalities are in the form ax + b > c, where a, b, and c are constants and x is a variable.

Question 2

How do you solve a linear inequality?

Accepted Answer

To solve a linear inequality: 1) Isolate the variable term (move constants to the other side), 2) Divide both sides by the coefficient, 3) If dividing by a negative number, FLIP the inequality sign, 4) Simplify to get the solution. The process is similar to solving equations, but you must flip the sign when multiplying/dividing by negative numbers.

Question 3

Why do you flip the inequality sign?

Accepted Answer

You flip the inequality sign when multiplying or dividing both sides by a negative number because the direction of the inequality reverses. For example: -2x > 6 becomes x < -3 after dividing by -2 and flipping the sign. This maintains the truth of the inequality statement.

Question 4

What is interval notation?

Accepted Answer

Interval notation is a way to represent solution sets using brackets. For x > 2: (2, ∞). For x ≥ 2: [2, ∞). Parentheses ( ) mean 'not including' (open), brackets [ ] mean 'including' (closed). Infinity always uses parentheses: (-∞, 2) or (2, ∞).

Question 5

How do you graph a linear inequality on a number line?

Accepted Answer

1) Identify the critical point (boundary value), 2) Use an open circle (○) for strict inequalities (>, <) or closed circle (●) for inclusive inequalities (≥, ≤), 3) Shade the side that satisfies the inequality. For x > 2, put an open circle at 2 and shade to the right.

Question 6

What's the difference between solving equations and inequalities?

Accepted Answer

The main differences: 1) Inequalities have multiple solutions (a range), while equations typically have one solution, 2) When multiplying/dividing by negatives, inequalities require flipping the sign, 3) Inequalities are graphed as rays or line segments on a number line, not just points.

Question 7

How do you solve compound inequalities?

Accepted Answer

Compound inequalities combine two inequalities with AND or OR. For AND: find the intersection (overlap) of both solution sets. For OR: find the union (all solutions that satisfy either inequality). Our compound inequality calculator handles these cases.

Question 8

What are some common mistakes when solving inequalities?

Accepted Answer

Common mistakes: 1) Forgetting to flip the sign when multiplying/dividing by negatives, 2) Not properly isolating the variable, 3) Confusing open and closed intervals, 4) Incorrectly graphing on the number line, 5) Mixing up the direction of shading. Always check your work by testing a value in the solution set.

Question 9

How are linear inequalities used in real life?

Accepted Answer

Linear inequalities model real-world situations with constraints: budget limitations, time constraints, capacity limits, safety thresholds, and optimization problems. Examples: x ≤ 500 (budget), t > 0 (time must be positive), speed < 65 mph (speed limit), profit ≥ 1000 (minimum profit requirement).

Question 10

Can linear inequalities have no solution or infinite solutions?

Accepted Answer

Yes. Some inequalities have no solution (like x > 5 and x < 3 simultaneously - impossible). Others have all real numbers as solutions (like x + 1 > x, which simplifies to 1 > 0, always true). Most have a solution set that's a ray or half-line.

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Inequality: ax + b > c

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