What Are They?
**Inequalities** are mathematical sentences that compare two values using symbols like `>` (greater than), `<` (less than), `≥` (greater than or equal to), and `≤` (less than or equal to). The solution to an inequality is a range of numbers.
**Absolute value** represents the distance of a number from zero on the number line. It's always a positive value, denoted by two vertical bars: $|x|$. For example, $|-5| = 5$ and $|5| = 5$.
To solve absolute value equations like $|ax+b|=c$, you must consider two cases: $ax+b=c$ and $ax+b=-c$.
Video: Solving Absolute Value Equations 🎬
This video will show you how to solve an absolute value equation by considering two cases.
Applications & Modeling
Both inequalities and absolute value are used to describe real-world constraints and tolerances. They're everywhere, from engineering to daily life.
For example, a car's speed on a highway can be described with an inequality. If the speed limit is 70 mph and the minimum speed is 60 mph, your speed $s$ must satisfy the inequality: $$60 \le s \le 70$$
Absolute value is often used to model a margin of error or a deviation from a target value. A machine part with a target length of $10 \text{cm}$ that has a tolerance of $\pm 0.1 \text{cm}$ can be described by the absolute value inequality: $$|L - 10| \le 0.1$$
Practice Problems
Solve the following problems and check your answers.
$$2x + 1 > 5$$
Answer:
$$|x-3| = 4$$
Answer:
$$|-15|$$
Value:
$$3x - 5 \le 10$$
Value:
⚡ Speed Quiz
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