MATH 110: course materials - operations for equations and inequalities

COLLEGE ALGEBRA MATH 110		Southwestern Adventist University
Distance Education	Lawrence E. Turner, Jr., Ph.D.

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Equations and Inequalities

There are several specific operations that are permitted in manipulating the two sides of an equation or an inequality. Since most students have more experience with equations, these can be rather familiar. The tendancy is to handle inequalities exactly like equations. However, while these operations are generally the same there are some subtle but important differences.

Generally, it is helpful to realize that an equation or inequality is a statement — a True statement. Whatever we do to it to change it, we still want the result to also be True. By performing a series of operations to a True statement, each which leaves it True, then we can manipulate it to solve for a variable, simplify, etc. with confidence that the final result is still True.

We start with an equality a = b and an inequality a < b. Here the less than comparison operator could be any of the four: less than (<), less than or equal to (≤), greater than (>), greater than or equal to (≥). The principle is the same.

Now let us perform some operation such as adding (see the first row below) the same non-zero constant, c to each side. If we start with a True statement, then we want to end up with a True statement.

	equation a = b	inequality a < b
addition	a+c = b+c		a+c < b+c
subtraction	a−c = b−c		a−c < b−c
multiplication	ac = bc	c>0	ac < bc
multiplication	ac = bc	c<0	ac > bc
division	a/c = b/c	c>0	a/c < b/c
division	a/c = b/c	c<0	a/c > b/c
reverse	b = a		b > a

The last row is rather obvious. Exchanging the two sides of a True equation still results in a True equation. Clearly when we interchange the two sides of an inequality, it would make sense that we need to reverse the sense of the comparison operator.

When we multiply or divide the two sides of an inequality by a negative value, then we must reverse the sense of the inequality operator. For an equality, it does not make any difference.

Thus the rules for dealing with a inequality are much the same as handling an equality; however, when multiplying or dividing both sides, it is necessary to make an additional mental check: if the value is negative, then we must also reverse the sense of the comparison operator that links the two sides together.