SAT-Math-13: Solving Multi-Step Linear Inequalities

1. Inequalities vs. Equations

Solving an inequality is almost identical to solving a regular linear equation. You use the exact same inverse operations to isolate your variable.

However, instead of an equal sign ($=$), you work with four specific inequality symbols:

• $<$ : Less than (kichik)
• $>$ : Greater than (katta)
• $\le$ : Less than or equal to (kichik yoki teng)
• $\ge$ : Greater than or equal to (katta yoki teng)

2. The Golden Rule: The Negative Flip Trap

There is one massive trap that the SAT sets on almost every single inequality question.

⚠️ The Absolute Golden Rule: Whenever you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign.

• If you have $-2x < 10$ and divide by $-2$, the sign flips: $x > -5$.
• If you multiply or divide by a positive number, the sign stays exactly the same.

🇺🇿 Uzbek Explanation:

Tengsizliklarning Oltin Qoidasi: Tengsizlikni yechish oddiy tenglamani yechish bilan bir xil. Ammo bitta juda muhim qoida bor: tengsizlikning ikkala tomonini manfiy songa ko'paytirganda yoki bo'lganda, tengsizlik ishorasi teskari tomonga o'zgaradi (masalan, $<$ ishora $>$ ga aylanadi). Musbat songa bo'lganda esa ishora o'zgarmaydi.

3. High-Yield SAT Practice Question

Question: Which of the following represents all possible values of $x$ that satisfy the inequality $5 - 3(x - 2) \ge 20$?

A) $x \ge -3$

B) $x \le -3$

C) $x \le -5$

D) $x \ge -5$

Step-by-Step Explanation:

1. Distribute the $-3$ safely across the parenthesis:

   $$5 - 3x + 6 \ge 20$$

2. Combine like terms on the left side ($5 + 6 = 11$):

   $$11 - 3x \ge 20$$

3. Isolate the variable term by subtracting $11$ from both sides:

   $$-3x \ge 9$$

4. Divide by $-3$: Because we are dividing by a negative number, we must flip the sign from $\ge$ to $\le$:

   $$x \le \frac{9}{-3}$$

   $$x \le -3$$

Correct Answer: B

Summary

• Treat inequalities like equations, but maintain the orientation of the symbol.
• Flip the symbol instantly anytime you multiply or divide the entire expression by a negative value.
• Always double-check your arithmetic signs during the distribution step to avoid missing a double negative.