When solving a compound inequality that uses "and", what is the recommended method?

When solving a compound inequality that uses "and", what is the recommended method?

• A. Solve each part separately and find the overlap
• B. Solve only the larger interval
• C. Swap the inequality signs
• D. Convert to a system of equations

Answer: (A)

When a compound inequality uses “and,” you’re looking for values that satisfy both conditions at once. Solve each part separately to get its solution set, then find where those sets overlap. That overlap—the intersection—gives the final solution because only those x-values make both inequalities true. Think of it on a number line: shade the region for each inequality. The solution is where the shaded regions coincide. For example, x > 1 and x < 4 gives regions (1, ∞) and (-∞, 4); their overlap is (1, 4), so those x work for both. Avoid thinking you should pick the larger interval or convert to a system of equations—that would ignore the requirement that both conditions must hold.

When a compound inequality uses “and,” you’re looking for values that satisfy both conditions at once. Solve each part separately to get its solution set, then find where those sets overlap. That overlap—the intersection—gives the final solution because only those x-values make both inequalities true.

Think of it on a number line: shade the region for each inequality. The solution is where the shaded regions coincide. For example, x > 1 and x < 4 gives regions (1, ∞) and (-∞, 4); their overlap is (1, 4), so those x work for both.

Avoid thinking you should pick the larger interval or convert to a system of equations—that would ignore the requirement that both conditions must hold.