Linear Inequalities

An inequality is solved the same way as an equation, by doing the same thing to both sides. For three x plus five is less than twenty, subtract five from both sides: three x is less than fifteen. Divide both sides by three: x is less than five. The answer is not one number but a whole range: every number below five. Test one: x equals four gives seventeen, and seventeen is less than twenty. On the number line, draw an open circle at five, because five itself is not included, and shade everything to the left.

Here is the one rule that is different from equations. Two is less than five. Multiply both by negative one and you get negative two and negative five, but now negative two is the bigger one. The order has flipped. So whenever you multiply or divide both sides by a negative number, you must reverse the inequality sign. For negative two x is greater than six, divide both sides by negative two and reverse: x is less than negative three. Test it: x equals negative four gives eight, and eight is greater than six.

With x on both sides, you can avoid dividing by a negative altogether. In seven minus two x is greater than or equal to three x minus eight, add two x to both sides: seven is at least five x minus eight. Add eight: fifteen is at least five x. Divide by five, which is positive, so no flip: three is at least x. Read that from the x's point of view: x is less than or equal to three. If you had gathered the x's on the left instead, you would get negative five x is at least negative fifteen, and dividing by negative five with the sign reversed gives the same answer.

A range with two ends, like negative two is less than or equal to x, which is less than three, is drawn with a circle at each end. Negative two has a closed circle, because it is included. Three has an open circle, because it is left out. The integers in the range are negative two, negative one, zero, one and two. That is five integers. Three is not on the list, because of its open circle.

A compound inequality has three parts, like negative three is less than two x plus one, which is less than or equal to nine. Do the same thing to all three parts. Subtract one: negative four is less than two x, which is at most eight. Divide all three parts by two: negative two is less than x, which is at most four. So x is between negative two and four, with negative two left out and four included. The integers are negative one, zero, one, two, three and four. If you ever divide all three parts by a negative, reverse both signs. Quiz and exercises now.