Overview: What Are Inequalities?
Most of the time, in order to solve problems in pre-algebra or algebra, students use equations, where number sentences are exactly equal. However, some number sentences use expressions that are inequalities, such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). They are solved by following similar rules to equations, with a few differences.
Addition and Subtraction Rules for Inequalities
Addition rules for inequalities are very similar to addition rules for equations. The commutative property for addition holds in a special way. If a>b , then a +c > b +c, and a – c> b – c. (Remember that letters stand for the same number or both sides of the sentence. If a =5 , b=4, and c =1, then 5>4, and 5+1 > 4+1. The letters don’t stand for something else in the middle of the problem.
Is “Less than” Similar to “Greater than”?
The rules for less than are similar to the rules for greater than. If a <b, then a +c < b +c and a-c < b-c.
For example, the number sentence is x-7≥-8. Solving for x and using the rules, get x as the only number to the right of the inequality the same way as solving an equation, by adding 7 to each side. Therefore, x-7 +7 ≥ -8 +7, or x-0 ≥ -1, or x.≥-1.
Multiplication and Division Rules for Inequalities
Multiplication (and its inverse, division) have similar rules to addition and subtraction. For all numbers a, b, c , if a >b, then a times c > b times c, and a/c > b/c. If a<b, then a(c) < b(c), and a/c < b/c. To test that in the real world, let a =5, b=4, and c =1, but it will work for any positive real number. For example, 5 >4, so 5(1) > 4(1); and 5/1 > 4/1.
If c is negative, the sign is reversed, because of the rules of multiplication by negative numbers. Therefore, for all numbers a, b, and c, if c is negative, if a >b, than a(c) <b(c), and a/c<b/c. This time, let a =5, b =4, and c=-2. So, 5>4, but 5(-2) or -10 < 4(-2) or -8. That is because the larger the negative number is, the further away it is from zero in the opposite direction.
Graphing Inequalities
A linear equation will graph as a straight line and will divide the coordinate plane into two regions, one above the line and one below. The only solutions to the equation will fall on the line. However, an inequality will graph as a portion of the coordinate plane. The solutions to the inequality will fall just below the line if the number sentence is < (less than) or > (greater than). They will include the line if the number sentence is ≤ (less than or equal to) or ≥ (greater than or equal to).
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