2.7 Linear Inequalities and Absolute Value Inequalities

Linear inequalities compare expressions using symbols like < or ≥. They're solved by isolating variables and graphed on number lines. Understanding interval notation and inequality properties is key to finding and representing solution sets.

Absolute value inequalities involve the distance from zero. They're rewritten as compound inequalities and solved similarly to linear ones. Mastering these concepts helps tackle more complex algebraic problems and real-world applications.

Linear Inequalities

Interval notation for inequalities

• Represents the solution set of an inequality using parentheses ( ) for exclusive endpoints (strict inequalities) and brackets [ ] for inclusive endpoints (non-strict inequalities)
• Uses $\infty$ to represent infinity and $-\infty$ for negative infinity
• Expresses $x < 5$ as $(-\infty, 5)$, $x \geq -2$ as $[-2, \infty)$, and $-3 < x \leq 4$ as $(-3, 4]$

Properties of inequality solutions

• Maintains inequality when adding or subtracting the same value on both sides ($x < 5$, then $x + 3 < 5 + 3$ or $x + 3 < 8$)
• Preserves inequality when multiplying or dividing both sides by a positive number ($2x > 10$, then $x > 5$)
• Reverses inequality when multiplying or dividing both sides by a negative number ($-3x < 9$, then $x > -3$)
• Involves algebraic manipulation to isolate the variable and solve for the inequality

One-variable inequality graphing

• Isolates the variable on one side using inequality properties ($2x - 5 < 7$ becomes $2x < 12$, then $x < 6$)
• Represents strict inequalities (< or >) with an open circle (○) and non-strict inequalities ($\leq$ or $\geq$) with a closed circle (●) on a number line
• Shades the portion of the number line satisfying the inequality (for $x < 6$, open circle at 6 and shade left)
• Graphing inequalities visually represents the solution set on a number line

Absolute Value Inequalities

Absolute value inequality interpretation

• Rewrites absolute value inequalities as compound inequalities:
  1. $|x| < a$ as $-a < x < a$
  2. $|x| > a$ as $x < -a$ or $x > a$
  3. $|x| \leq a$ as $-a \leq x \leq a$
  4. $|x| \geq a$ as $x \leq -a$ or $x \geq a$
• Solves $|2x - 3| < 5$ by rewriting as $-5 < 2x - 3 < 5$, adding 3 to all parts ($-2 < 2x < 8$), and dividing by 2 ($-1 < x < 4$)

Understanding Inequalities

• An inequality is a mathematical statement that compares two expressions using inequality symbols (<, >, ≤, ≥)
• Inequalities involve variables, which represent unknown values or quantities in mathematical expressions
• Solving inequalities requires understanding how to manipulate expressions while maintaining the relationship between the two sides