. Asolte Vale Eqations mer of asolte vales 3 3= o soltion Isolate the asolte vale Other side negative? Rewrite the eqation with one asolte vale on each side Write two eqations withot asolte vales: In one, simply omit the asolte vales In the other, omit the asolte vales and negate one side Solve each eqation 3 = 3 = or 3 = 3 = 6 or 3 = = or = 3. Asolte Vale Ineqalities Isolate the asolte vale on the left > < Which ineqality symol? Other side Other side negative? negative? The soltion is all real nmers Solve compond ineqality with OR o soltion Solve compond ineqality with AD See eample elow See eample elow 3 > 7 3> 7 or 3< 7 > or < ) ( - 3 < 7 7< 3< 7 < < ( )
3. Polynomial Eqations Move all terms to the same side of the eqation and place them in descending order Factor the reslting polynomial Set each factor eqal to zero and solve the reslting eqations 3 = 3 = ( 6)( + ) = ( ) = = or 6= or + = = or = 6 or =. Fractional Eqations 6 3 = 3 8 Any denominator with a variale? 6 3 = 3 8 ( 3) Write down all vales which the variale cannot have 3 Mltiply oth sides of the eqation y the LCD to clear all fractions 6 3 8( 3)( ) (8)( 3) 3 8 = ( 3) 8(6) 3( 3) = () 8 3 + 9 = Solve the reslting eqation eliminate any vales which the variale cannot have 7 3 = 3 = =
. Radical Eqations Isolate the radical mer of radicals? Are the indices the same?. + + 8 = + = 7 ( + ) = 7 + = 9 = Isolate the glier radical Go to #7: Eqations in qadratic form. 3+ + = 3+ = Raise oth sides to the power that matches the inde Comine like terms ( 3+ ) = ( ) 3+ = + 3 = + Any remaining radicals? = = Solve the reslting eqation ( ) = ( ) = = ( ) = = or = = or = Be sre to CHECK each soltion in the original eqation! OL = works in the original eqation!
6. Qadratic Eqations Move all terms to the same side Comine like terms. 6 + = + + 36 = = 36 Only one term with a variale? = 9 = ± 9 Isolate that term Solve y taking the sqare roots = ± 3 See eample Any fractions?. + 3 + = + 3 Mltiply all terms y the LCD Will it factor? Use the qadratic formla See eample Factor Set each factor eqal to zero and solve the reslting eqations ± = 8+ = ac a a = = 8 c = 8 ± 6 ()() = = = 8 ± 6 8 ± = ±
7. Eqations in Qadratic Form These are eqations that may e written in the form a( ) + ( ) + c = where a,, and c are nmers and where the parentheses may contain any algeraic epression. Some eamples are: (3+ ) + (3+ ) + = 3 + = or ( ) 3( ) + = 6 + = or 3 6( ) + ( ) = 3 + = or ( ) ( ) + = (3+ ) + (3+ ) + = Give a name to the algeraic epression within parentheses, say Let = 3+ Use this name to rewrite the eqation as a + + c = Then + + = Solve this qadratic eqation to find For each vale of otained, write an eqation sing the epression within parentheses from the original eqation: = ( ) Solve each eqation for the variale within the parentheses ( + )( + ) = + = or + = = or = Since = 3+ 3 + = or 3 + = 3 = or 3 = = or = 3 3
8. Eponential Eqations Sccess Center Math Tips. + 9 = 7 + 3 (3 ) = (3 ) Can the ases e rewritten as powers of the same nmer? + = 3 = + +. 6 = Rewrite the eqation sing the same ase on oth sides Eqate the eponents and solve for the nknown See eample 9. Logarithmic Eqations Take the log of each side Use the power rle for logs to ring down the eponents: r log = rlog Solve for the nknown See eample log 6 = log + + (+ )log 6 = ( + )log log 6 + log 6 = log + log log 6 log = log log 6 (log 6 log) = log log 6 log log 6 = log 6 log.79 log = + log( ) Move all terms with a log to one side of the eqation and all terms withot a log to the other side Use the rles for logarithms to rewrite the side with all the logs as a single log: logv = log + logv log log logv v = r log = rlog Rewrite the reslting eqation in eponential form and solve log log( ) = log = = = ( ) = = 99 = = 99