Lesson Solving Linear Inequalities
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Algebra: Inequalities, trichotomy
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The process for solving inequalities is similar to the process for solving equations. Just as with equations, you can add or subtract the same number from both sides of an inequality: <table border="1" cellspacing="0" cellpadding="0"> <tr> <td width="146" valign="top"> </td> <td width="118" valign="top"> EQUATION </td> <td width="132" valign="top"> INEQUALITY 1 </td> <td width="180" valign="top"> INEQUALITY 2 </td> </tr> <tr> <td width="146" valign="top"> Original: </td> <td width="118" valign="top"> X + 3 = 9 </td> <td width="132" valign="top"> X + 3 > 9 </td> <td width="180" valign="top"> X + 3 < 9 </td> </tr> <tr> <td width="146" valign="top"> Subtract 3 from both sides: </td> <td width="118" valign="top"> X = 6 </td> <td width="132" valign="top"> X > 6 </td> <td width="180" valign="top"> X < 6 </td> </tr> </table> And you can multiply or divide both sides of an inequality by a positive number: <table border="1" cellspacing="0" cellpadding="0"> <tr> <td width="146" valign="top"> </td> <td width="118" valign="top"> EQUATION </td> <td width="132" valign="top"> INEQUALITY 1 </td> <td width="180" valign="top"> INEQUALITY 2 </td> </tr> <tr> <td width="146" valign="top"> Original: </td> <td width="118" valign="top"> 5x = 10 </td> <td width="132" valign="top"> 5x > 10 </td> <td width="180" valign="top"> 5X < 10 </td> </tr> <tr> <td width="146" valign="top"> Divide both sides by 5: </td> <td width="118" valign="top"> X = 2 </td> <td width="132" valign="top"> X > 2 </td> <td width="180" valign="top"> X < 2 </td> </tr> </table> However, when you multiply or divide an inequality by a negative number, <i>the direction of the inequality reverses:</i> <table border="1" cellspacing="0" cellpadding="0"> <tr> <td width="146" valign="top"> </td> <td width="118" valign="top"> EQUATION </td> <td width="132" valign="top"> INEQUALITY 1 </td> <td width="180" valign="top"> INEQUALITY 2 </td> </tr> <tr> <td width="146" valign="top"> Original: </td> <td width="118" valign="top"> -5X = 10 </td> <td width="132" valign="top"> -5X > 10 </td> <td width="180" valign="top"> -5x < 10 </td> </tr> <tr> <td width="146" valign="top"> Divide both sides by - 5: </td> <td width="118" valign="top"> X = -2 </td> <td width="132" valign="top"> X < -2 </td> <td width="180" valign="top"> X > -2 </td> </tr> </table> Example. Solve for x: -3x + 5 < -7 Subtract 5 from both sides: -3x < -12 Divide both sides by –3: x > 4 (Note: the inequality symbol switches directions!) Try a sample problem <a href = "http://www.mathick.com/mathick.php?topic=ineq1">here</a>.
