Sample cheat sheet for Math V03 Examination 1 Recipe #1: Solving equations and inequalities (containing NO absolute values) Recipe #3: Absolute value equations (“LHS” = left hand side; “RHS” = right hand side) 1. If one or more fractions are present, find the LCD using both sides of the equation/inequality. Always use a positive LCD, even if some of the fractions are negative. If no fractions are present, skip to step 5. 2. Parenthesize each fraction or term, and multiply each by (LCD/1) or (LCD). 3. Cross-cancel (do not distribute yet) to remove the denominators. 4. If any denominators remain (other than some 1s), your “LCD” was incorrect. Instead of recalculating it, repeat steps 1 through 3 with the new fraction(s). 5. Apply order of operations to the remaining parentheses. Often this means distributing the remaining parentheses, taking special care on the ones preceded by minus signs or negative numbers to distribute the minus or negative to all terms in the parentheses. 6. Combine like terms. 7. Start to isolate the variable by adding or subtracting terms to/from both sides of the equation/inequality until there is only one term on each side of the equation. 8. If a coefficient remains in front of the variable to be isolated, divide both sides of the equation/inequality by this coefficient. If the coefficient is a fraction, multiply both sides of the equation/inequality by the reciprocal of this fraction instead. (“Reciprocal” means “turn the fraction upside-down.”) 9. (For inequalities only) If this last coefficient was negative, reverse the inequality symbol. 1. If the equation contains any expressions inside absolute Recipe #2: Isolating the variable in “for” problems (“LHS” = left hand side; “RHS” = right hand side) 1. Apply steps 1 through 6 from Recipe #1, so the formula contains no fractions, no parentheses, and no uncombined like terms before continuing. 2. If the “for” variable occurs more than once anywhere in the formula, then add/subtract terms to/from both sides so that all the terms containing the “for” variable end up on the LHS, and all terms NOT containing the “for” variable end up on the RHS (or vice-versa). 3. Combine like terms on both sides, if there are any. 4. If the “for” variable appears on the RHS, then reverse the equation (for example, change a b to b a ). 5. If the “for” variable occurs more than once on the LHS, then factor the LHS into a distributive expression, placing the “for” variable outside the parentheses. 6. Divide both sides (or multiply by a reciprocal) as needed to complete the isolation of the “for” variable. values that are constants only, such as 8 or 3 5 , 2. 3. 4. 5. simplify these expressions using order of operations. If both sides of the equation are totally enclosed in absolute value symbols, then drop the symbols on the simpler-looking side of the equation. If not already so, reverse the equation so the absolute value expression is on the LHS; parenthesize the RHS. If there are quantities outside the absolute values on the LHS of the equation, then isolate the absolute value expression by removing these quantities; add, subtract, divide, or multiply on both sides as needed. If the number on the RHS is negative, STOP, write “no solution” , and proceed no further with this recipe. 6. Split the equation into two halves: expression inside stripes RHS OR expression inside stripes RHS 7. Apply Recipe #1 to solve both equations in parallel. 8. If the two solutions are different numbers, write using “OR.” For example, x 4 OR x 10 . You may also use set notation, for example, 4,10 . Do NOT use parentheses or brackets; they refer to intervals. Recipe #4: Solving “OR” inequalities 1. Apply Recipe #1 to solve both inequalities in parallel. 2. Draw three number lines in a vertical column. Place tick marks representing the solution values found in step 1 on all three number lines in the same way. For example, if the solutions were x 4 OR x 7 , place tick marks at 4 and 7 on all three number lines. Make sure the marks align vertically for all three lines, and that the smaller solution value is farther to the left. 3. Shade one inequality’s solution on the first number line, and the other inequality’s solution on the second one. 4. For each of the three segments (“chunks”) of the third number line, shade only if the corresponding segment of either the first number line or the second number line (or both) directly above is shaded. Do not shade a segment on the third number line if neither of the first two number lines is shaded directly above it. 5. Looking only at the third number line (not the original problem or the first two number lines), write interval notation for the region(s) which are shaded. You may write , or if the entire third line is shaded. Use the union symbol “ ” if two or more separate segments are shaded on the third number line. Recipe #5: Solving “AND” inequalities 1. Apply steps 1 through 3 of Recipe #4. 2. For each of the three segments (“chunks”) of the third number line, shade only if the corresponding segments of both the first number line AND the second number line directly above are shaded. Do not shade a segment on the third number line if only one (or neither) of the first two number lines is shaded for that segment. 3. Looking only at the third number line (not the original problem or the first two number lines), write interval notation for the region which is shaded. You may write if no segment of the third line is shaded. If only a single number is shaded (in other words, if the solution is a single dot on the number line), use set notation, such as 3 , not interval notation such as 3, 3 . Recipe #6: Solving compound inequalities 1. Apply steps 1 through 6 of Recipe #1 to all three sides. 2. Start to isolate the variable (in the middle “side” only) by adding or subtracting terms to/from both sides of the inequality until there is only one term on all three sides of the inequality. 3. If a coefficient remains in front of the variable to be isolated, divide both sides of the inequality by this coefficient. If the coefficient is a fraction, multiply both sides of the inequality by the reciprocal of this fraction instead. (“Reciprocal” means “turn the fraction upside-down.”) 4. If this last coefficient was negative, reverse the inequality symbols. 5. If applicable, reverse any equality containing > or (such as 5 x 3 ) to a form containing < or (such as 3 x 5 ). The inequality symbols MUST face the same direction; if it looks like 3 x 5 or 3 x 5 then it is meaningless and therefore illegal. 6. Graph on a number line and/or write the solution using interval notation as directed. Additional notes Recipe #7: Absolute value inequalities, “>” or “ ” (“LHS” = left hand side; “RHS” = right hand side) 1. If the inequality contains any expressions inside absolute values that are constants only (no variables), such as 8 or 3 5 , then simplify these expressions using order of operations rules. 2. If not already so, reverse the inequality so the absolute value expression is on the LHS. 3. If there are quantities outside the absolute values on the LHS of the inequality, then isolate the absolute value expression by removing these quantities; add, subtract, divide, or multiply on both sides as needed. 4. If the number on the RHS is negative, STOP, write , or (the solution is all real numbers), and proceed no further with this recipe. 5. Split the inequality into two halves: expression inside stripes RHS OR expression inside stripes RHS , or use the and symbols if the original problem contained . IMPORTANT: The symbol in the negative half is always reversed to < or (this is not a typo). 6. Apply Recipe #1 to solve both inequalities in parallel. 7. Apply steps 2 through 5 of Recipe #4 to graph the solution and write it in interval notation. Recipe #8: Absolute value inequalities, “<” or “ ” (“LHS” = left hand side; “RHS” = right hand side) 1. Apply steps 1 through 3 of Recipe #7 to put the inequality into a workable form. 2. If the number on the RHS is negative, STOP, write “no solution” , and proceed no further with this recipe. 3. Create a compound inequality of the form RHS expression inside stripes RHS , or use the symbol if the original problem contained . 4. Apply Recipe #6 to solve the compound inequality. Additional notes