What is the fundamental difference between solving $2x = 10$ and $-2x < 10$?

In the equation, dividing by $-2$ simply gives $x = -5$. In the inequality, dividing by a negative number requires flipping the sign, resulting in $x > -5$.

When should you use an open circle versus a closed circle on a number line?

Use an open circle for strict inequalities ($ $) to show the boundary is excluded. Use a closed circle for non-strict inequalities ($\le$ or $\ge$) to show the boundary is included.

Why is it dangerous to multiply both sides of an inequality by a variable like $x$?

Because the value of $x$ could be negative, which would require flipping the sign. If you don't know the sign of $x$, you cannot determine the correct direction of the resulting inequality.

What happens to the inequality sign if you subtract a large positive number from both sides?

Nothing happens to the sign. Addition and subtraction never change the direction of an inequality, regardless of the sign or magnitude of the number being moved.

Define a 'strict inequality' and provide its symbols.

A strict inequality is one where the two sides cannot be equal. The symbols used are 'less than' ($ $).

If a question asks for 'integers' satisfying $2.1 < x < 4.9$, what are the solutions?

The integers are $3$ and $4$. Integers are whole numbers, so you must find the whole numbers that fall strictly between the two decimal boundaries.

What is the most common mistake when solving an inequality like $5 - x > 10$?

The most common mistake is forgetting to flip the sign when dividing by $-1$ to isolate $x$. The correct steps lead to $-x > 5$, which becomes $x < -5$.

How do you verify if your solution to an inequality is correct without looking at an answer key?

Pick a 'test value' that falls within your solution range and substitute it into the original inequality. If the resulting statement is true, your solution is likely correct.

Compare the solution sets of $x > 5$ and $x \ge 5$.

The set $x > 5$ includes all values greater than 5 but not 5 itself (e.g., 5.01, 6). The set $x \ge 5$ includes all those values plus the number 5.

What error is made if a student writes $x = 4$ as the answer to $x < 5$?

The student has treated the inequality as an equation. $x < 5$ represents an infinite range of values, not a single point, and should be left as an inequality unless specific integers are requested.

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Algebra

Solving Linear Inequalities

Summary

Linear inequalities represent relationships where one algebraic expression is greater than or less than another. Solving them involves isolating the variable using algebraic operations similar to equations, with the critical distinction that multiplying or dividing by a negative number reverses the inequality's direction.

1. Definition & Core Concepts

Linear Inequalities are mathematical statements that compare two linear expressions using inequality symbols: greater than ( $>$ ), less than ( $<$ ), greater than or equal to ( $\ge$ ), and less than or equal to ( $\le$ ).
A strict inequality ( $>$ or $<$ ) indicates that the boundary value is not included in the solution set, whereas a non-strict inequality ( $\ge$ or $\le$ ) includes the boundary.
Unlike linear equations which typically have one specific solution, linear inequalities usually result in a range of values or an infinite set of solutions that satisfy the condition.
Solutions can be restricted to integers (whole numbers, including zero and negatives) or include all real numbers (decimals, fractions, and irrational numbers) depending on the problem constraints.

Number line visualization showing a closed circle for non-strict inequalities (inclusive) and an open circle for strict inequalities (exclusive).

2. The Negative Rule (Underlying Principle)

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions