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