AI Tutor
If your phone battery needs to stay above 20% to avoid low-power mode, you do not have just one target percentage — you have an entire range of acceptable charge levels.
A linear inequality is a mathematical statement comparing two linear expressions using order relations like <<, >>, ≤\le, or ≥\ge.
Solving an inequality feels almost identical to solving an equation, until one crucial operation completely flips the rules.

The Negative Multiplier Rule

According to the Sign Reversal Property, multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol (e.g., << becomes >>, ≤\le becomes ≥\ge) because it reflects positions across zero on the number line.
Consider the true statement 2<52 < 5. If you multiply both sides by −1-1, you get −2-2 and −5-5; because −2-2 sits to the right of −5-5 on the number line, −2>−5-2 > -5.
How do we apply this rule when an inequality has grouping symbols, fractions, and variables on both sides?

Solving Multi-Step Inequalities

To solve a multi-step inequality, first apply the distributive property to eliminate parentheses, combine like terms, and then isolate the variable.
Let's solve 2(3x−1)−8x≤102(3x - 1) - 8x \le 10. Distribute the 2 to get 6x−2−8x≤106x - 2 - 8x \le 10, and combine like terms to reach −2x−2≤10-2x - 2 \le 10.
Add 22 to both sides to get −2x≤12-2x \le 12. Finally, divide by −2-2 and flip the inequality symbol to obtain x≥−6x \ge -6.