lesson 6: Equations and Inequalities




lesson 6

Equations and Inequalities

Rules for Equality: For all real numbers a, b, and c, if a = b, then the
following rules are valid.

$${\rm{a + c = b + c}}$$
$${\rm{a – c = b – c}}$$
$${\rm{a*c = b*c}}$$
$${a \over c}{\rm{ = }}{b \over c}\;whenever\;c \ne 0$$

In other words, we can perform the four basic operations and preserve equality as long as we do the same thing to both sides.

Example: Solve for x:

$${\rm{x + 5 = 9}}$$

We would like to change x + 5 to x, so we will subtract 5 from both sides.

$$\eqalign{
& {\rm{x + 5 – 5 = 9 – 5}} \cr
& {\rm{x = 4}} \cr} $$

Inequality

Just as an equation relates two expressions that are equal, an inequality
relates two expressions that are not equal. The symbols used in inequalities are as
follows:

$$\eqalign{
& > \;Greater\;than \cr
& < \;Less\;than \cr & \ge \;Greater\;than\;or\;equal \cr & \le \;Less\;than\;or\;equal \cr} $$

The inequality changes direction when both sides are multiplied or divided by a negative
number.

Example: Solve for x:

$$\eqalign{
& – 3x > 6 \cr
& {{ – 3x} \over { – 3}}\;?\;{6 \over { – 3}} \cr
& x < - 2\;{\rm{(notice}}\;{\rm{that}}\; > \;{\rm{became}}\;{\rm{ < )}} \cr} $$

Rules for Inequality: For all real numbers a, b, and c, if a < b, then the following rules are valid:

  1. $$a + c < b + c$$
  2. $$a – c < b - c$$
  3. $$a*c < b*c,\;whenever\;c > 0$$
  4. $$a*c > b*c,\;whenever\;c < 0$$
  5. $${a \over c} < {b \over c},\;whenever\;c > 0$$
  6. $${a \over c} > {b \over c},\;whenever\;c < 0$$

Example: Solve for x:

$$\eqalign{
& 3x + 2 < 5x + 8 \cr & 3x + 2 - 5x < 5x + 8 - 5x \cr & - 2x + 2 < 8 \cr & - 2x + 2 - 2 < 8 - 2 \cr & - 2x < 6 \cr & {{ - 2x} \over { - 2}} > {6 \over { – 2}},\;(notice\;that\; < \;became\; > ) \cr
& x > – 3 \cr} $$


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