|
 |
|||||
Solving absolute value inequalitiesWhen solving absolute value inequalities, the process is very similar to solving absolute value equations. You should review the latter before studying this lesson. Absolute value definition: If x is positive, | x | = x If x is negative, | x | = -x Example #1: Solve for x when | x | < 8 After applying the definition to example #1, you will have two equations to solve In fact, when solving absolute value inequalities, you will usually get two solutions. That is important to keep in mind If x is positive, | x | = x, so the first equation to solve is x < 8. Done because x is automatically isolated If x is negative, | x | = -x, so the second equation to solve is -x < 8. You can write -x < 8 as -1x < 8 and divide both sides by -1 to isolate x. (-1/-1)x > 8/-1 1x > 8/-1 x > -8 Notice that the smaller sign (<) was switched to a bigger sign (>) This happens whenever you divide or multiply inequalities by a negative number Let us see why this makes sense We know that 2 < 4. However, let us divide both sides by a negative number, say -2 2/-2 ? 4/-2 -1 ? -2 Should ? be < or >? Since -1 is bigger than -2, it should be -1 > -2 So, the solutions are x < 8 and x > -8 x > -8 means the same thing as -8 < x Putting -8 < x and x < 8 together, you can write -8 < x < 8 Therefore, any number between -8 and 8 is a solution for instance, if we choose -5, we get | -5 | = 5 and 5 is smaller than 8 Example #2: Solve for x when | x − 4 | < 7 Before, we apply the definition, let's make a useful substitution Let y = x − 4, so | x − 4 | < 7 becomes | y | < 7. You must understand this step. No excuses! Now, let's apply the definition to | y | < 7. Again, you will have two inequalities to solve Once again, when solving absolute value inequalities, you will usually get two solutions. If y is positive, | y | = y, so the first equation to solve is y < 7. No, you are not done! You have to substitute x − 4 for y After substitution, y < 7 becomes x − 4 < 7 x − 4 < 7 x + -4 < 7 x + -4 + 4 < 7 + 4 x < 11 If y is negative, | y | = -y, so the second equation to solve is -y < 7. You have to substitute x − 4 for y You get -( x − 4) < 7. Notice the inclusion of parenthesis this time -(x − 4) < 7 -(x + -4) < 7 -x + 4 < 7 -x + 4 − 4 < 7 − 4 -x < 3 (-1/-1)x > 3/-1 x > -3 The solutions are x > -3 and x < 11 Example #3: Solve for x when | 3x + 3 | > 15 Before, we apply the definition, let's make a useful substitution Let y = 3x + 3, so | 3x + 3 | > 15 becomes | y | > 15. Now, let's apply the definition to | y | > 15. Lastly, when solving absolute value inequalities, you will usually get two solutions. We may never say this enough! If y is positive, | y | = y, so the first equation to solve is y > 15. You have to substitute 3x + 3 for y After substitution, y > 7 becomes 3x + 3 > 15 3x + 3 > 15 3x + 3 − 3 > 15 − 3 3x > 12 (3/3)x > 12/3 x > 4 If y is negative, | y | = -y, so the second equation to solve is -y > 15. After substitution, -y > 7 becomes -(3x + 3) > 15 -(3x + 3) > 15 -3x + -3 > 15 -3x + -3 + 3 > 15 + 3 -3x > 18 (-3/-3)x > 18/-3 x < -6 The solutions are x > 4 and x < -6 Solving absolute value inequalities should be straightforward if you follow my guidelines above
 |
|
|||||
IntroductionArithmeticGeometryGraph it!Probability and statisticsReal life mathSequences and patternsAlgebraResourcesTeachers!
|
||||||
| Powered by Site Build It | ||||||
|
| ||||||
