Inequalities - Flipping the Sign, Examples

Main Ideas

• Inequalities are generally solved like equations: perform the same operation on both sides.
• Adding or subtracting the same amount from both sides preserves the inequality’s direction.
• When multiplying or dividing both sides by a negative number, reverse the inequality sign.
• Multiplying by a negative reverses the order of numbers. For example, (1<2), but multiplying both sides by (-3) gives (-3>-6). The inequality sign must be flipped to keep the statement true.

Examples and Checking

• If an inequality simplifies to (-5x>15), divide both sides by (-5) and reverse (>) to (<). The result is (x<-3). The subtitle says “divide both sides by 5,” but the stated answer requires dividing by (-5).

• To solve (-\frac{4x}{3}+5>6):

  1. Subtract 5: (-\frac{4x}{3}>1).
  2. Multiply by (-\frac{3}{4}), the reciprocal of (-\frac{4}{3}), and reverse the inequality sign.
  3. The result is (x<-\frac{3}{4}).

• To check a solution, substitute a value that should belong to the solution set into the original inequality. For example, if the result is (x<-3), test (x=-4). The substitution gives (3-5(-4)=23), and (23>18), supporting the inequality’s direction. A single check is useful but does not prove the entire solution is correct.

Speakers or Sources Featured

• Steve Stein — instructor and presenter; no other speakers are apparent in the subtitles.