SAT-Math-22: Absolute Value Inequalities on the Number Line

1. The Distance Framework on a Number Line

Just like absolute value equations, absolute value inequalities describe a geometric distance from a specific anchor point on a number line.

Depending on the direction of the inequality sign, the solution will form one of two distinct visual patterns: a bounded "sandwich" interval or a split "outer" interval.

Case 1: The "Less Than" Sandwich ($<$ or $\le$)

If an absolute value inequality uses a "less than" sign, it means the distance from zero must be smaller than a certain value. This creates a single, connected boundary line between a negative and a positive constraint.

$$\text{If } |x| \le 5 \implies -5 \le x \le 5$$

Case 2: The "Greater Than" Split ($>$ or $\ge$)

If an absolute value inequality uses a "greater than" sign, it means the distance from zero must be larger than a certain value. This splits your solutions into two completely separate arrows pointing outward in opposite directions.

$$\text{If } |x| \ge 5 \implies x \ge 5 \quad \text{or} \quad x \le -5$$

🇺🇿 Uzbek Explanation:

Modulli Tengsizliklar: Modul qatnashgan tengsizliklarni yechishda belgilarga juda ehtiyot bo'lish kerak:

• Kichik bo'lsa ($<$ yoki $\le$): Yechim o'rtada siqilib qoladi (Sandwich). Masalan, $|x| \le 5$ bo'lsa, javob $-5$ dan $5$ gacha bo'lgan oraliq bo'ladi (birlashtirilgan bitta oraliq).

• Katta bo'lsa ($>$ yoki $\ge$): Yechim ikki chetga qarab ajralib ketadi (Split). Masalan, $|x| \ge 5$ bo'lsa, javob $x \ge 5$ va $x \le -5$ kabi ikkita alohida cheksiz oraliq bo'ladi.

2. The Algebraic Solution Blueprint

To solve a multi-step absolute value inequality algebraically, isolate the absolute value bars first. Once isolated, split the expression based on the sign:

• For Less Than ($|\text{Bars}| < C$): Drop the bars and create a double inequality:

  $$-C < \text{Expression} < C$$

• For Greater Than ($|\text{Bars}| > C$): Drop the bars and create two separate inequalities joined by "or" (remember to flip the sign for the negative boundary):

  $$\text{Expression} > C \quad \text{or} \quad \text{Expression} < -C$$

3. High-Yield SAT Practice Question

Question: Which of the following inequalities represents all possible values of $x$ that satisfy $|2x - 3| \le 7$?

A) $-2 \le x \le 5$

B) $x \le 5$

C) $x \le -2 \quad \text{or} \quad x \ge 5$

D) $-5 \le x \le 2$

Step-by-Step Explanation:

1. Identify the setup: The absolute value bars are already isolated on the left, and the sign is $\le$ ("less than" $\implies$ Sandwich model).

2. Set up the double inequality:

   $$-7 \le 2x - 3 \le 7$$

3. Isolate $x$ by performing operations on all three sections of the inequality simultaneously.

   • Add $3$ everywhere:

     $$-7 + 3 \le 2x \le 7 + 3$$

     $$-4 \le 2x \le 10$$

   • Divide everything by $2$:

     $$\frac{-4}{2} \le x \le \frac{10}{2}$$

     $$-2 \le x \le 5$$

Correct Answer: A

Summary

• Isolate first: Never break open absolute value bars until they sit completely alone on one side of the inequality.

• Less Than $\implies$ Sandwich: Turns into a single combined statement ($-C \le \text{Expression} \le C$).

• Greater Than $\implies$ Split: Turns into two completely disjointed paths pointing outward ($\text{Expression} \ge C$ or $\text{Expression} \le -C$).

🎉 Congratulations, my bro! We have officially finished Block A: The Heart of Algebra (Lessons 1–22).