Identify the inequality type: linear, absolute value, rational, quadratic, or higher-degree
Simplify the inequality first (distribute, combine like terms, clear parentheses)
If fractions are present, multiply both sides by the least common denominator (flip the inequality if the multiplier is negative)
If absolute value is present, use the definition:
For ( |A| le c ): solve ( -c le A le c )
For ( |A| < c ): solve ( -c < A < c )
For ( |A| ge c ): solve ( A le -c ) or ( A ge c )
For ( |A| > c ): solve ( A < -c ) or ( A > c )
For linear inequalities, isolate the variable using inverse operations
When multiplying/dividing both sides by a variable expression, track its sign: split into cases if needed
For quadratic inequalities:
Move everything to one side to get ( ax^2+bx+c text{(relation)} 0 )
Find critical points (solve ( ax^2+bx+c=0 ))
Use a sign chart or test intervals to determine where the inequality holds
For rational inequalities:
Move to one side and factor if possible
Exclude values that make denominators zero
Use a sign chart on the factored expression
For systems of inequalities, solve each inequality and intersect the solution sets
Write the solution set in correct form:
Interval notation (e.g., ((a,b]))
Inequality form (e.g., (x le 3))
Number line description (open/closed circles)
Check the solution by substituting boundary values (respecting strict vs non-strict inequalities)
State the final solution set clearly (union for “or” conditions, intersection for “and” conditions)