Math 1 · Lesson 6 · part 4 of 6
The three steps, one rule added
Solve an inequality exactly the way Lesson 5 solved an equation.
The three steps, one rule added
Solve an inequality exactly the way Lesson 5 solved an equation. The properties have new last names (of inequality instead of of equality), and one of them has a warning label.
- 1RewriteTurn every subtraction into adding the additive inverse (the opposite), and every written ÷ sign into multiplying by the multiplicative inverse (the reciprocal). Write in every invisible 1: a lonely x is 1x, and −x is (−1)x.
- 2Distribute, then combine like termsHand the outside number to every term inside the parentheses (the distributive property), then combine the like terms (the terms with the same letter part) with the distributive property run backwards.
- 3x terms one side, plain numbers the otherAdd opposites to both sides (the addition property of inequality, the sign stays). Finish by multiplying both sides by the reciprocal of the coefficient (the multiplication property of inequality). If that reciprocal is negative, flip the sign.
- 4Check three numbersOne inside the answer, the boundary, and one outside. An inequality has many answers, so one number is not enough of a check.
Why the flip lands on the last stepThe only move that multiplies both sides is the last one, and it multiplies by the reciprocal of the coefficient (the number on the variable). A negative coefficient has a negative reciprocal. So look at the sign of the coefficient before you make that move: negative means the gator turns around. A tip from Lesson 5 helps here too: when x is on both sides, move the x term with the smaller coefficient, and the one you keep stays positive, so no flip is needed at all.