Inequalities

Inequalities show the relationship between quantities that aren't equal. Most inequalities have four different signs:

less than (<)

more than (>)

less or equal to ≤

greater than or equal to ≥

Normally, there are too sides of the inequality, and the signs (above) act the same as the equal sign when solving an inequality, but sometimes, there are more than two sides of an inequality. When solving inequalities with more than two sides, treat the sides the same as you would treat them if there were equal signs.

-Whatever you do to one side, do the other sides.

SOLVING INEQUALITIES

HOW TO SOLVE INEQUALITIES:

To solve inequalities is almost the same as solving equations, except for some things listed below this mini-section. First, we eliminate all constants and isolate the quantity/variable we are solving for, and for this, we'll be using the inequality [ 4x + 5 (≥ ) 37 ] To solve this, first we have to "destroy" the five, so we subtract 5 from both sides, turning the inequality into: 4x ≥ 32. Then, we divide both sides by 4, getting x ≥ 8.

Examples:

2x < 6: divide both sides by 2, getting x<3

7(3y+5) > 11*7: divide both sides by 7, getting 3y+5 > 11, subtract 5, 3y > 6, divide both sides by 3, getting y>6

NEGATIVE NUMBERS

When dividing or multiplying negative numbers from every side in any way, flip all signs. For example, when we have the equation -5x < 10, we solve it by dividing both sides by -5, but when we do that, we have to flip the sign, so -5x<10 -----------> x>-2. One thing to clarify is that if you don't change anything to more than one side, no sign needs to be flipped, ex: -5 * 4x < 40, we don't have to flip the sign when multiplying 4x by -5, since nothing happens to the other side (40), so -20x < 40, meaning x < -2.

Example:

-3(2t-7) < -63

divide both sides by -3 and reverse the sign, getting 2t-7 > 2
Add 7 to both sides, getting 2t > 9
Divide 2 from both sides, getting t > 4.5

ANSWER: t is greater than 4.5

GRAPHING INEQUALITIES

Quadratic Inequalities

While the previous sections cover how to solve and graph linear inequalities, quadratics can also reside in inequalities.

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