All GMAT Math Resources
Example Questions
Example Question #31 : Solving Inequalities
Give the solution set of the inequality
To solve a quadratic inequality, move all expressions to the left first:
The boundary points of the solution set will be the points at which:
: that is, , or
: that is, .
None of these values will be included in the solution set, since equality is not allowed by the inequality symbol.
Test the intervals
by choosing a value in each interval and testing the truth of the inequality.
: Test
True; include the interval .
: Test
False; exclude the interval .
: Test
True; include the interval .
The solution set is .
Example Question #32 : Solving Inequalities
Give the solution set of the inequality
To solve a rational inequality, move all expressions to the left first:
The boundary points of the solution set will be the points at which:
- that is, ;
; and
- that is, .
None of these values will be included in the solution set, since equality is not allowed by the inequality symbol.
Test the intervals
by choosing a value in each interval and testing the truth of the inequality.
- test
False - exclude
- test
True - include
- test
False - exclude
- test
True - include
The solution set is .
Example Question #33 : Solving Inequalities
Give the solution set of the inequality
The square of a real number must be nonnegative, so this is a true statement regardless of the value of . The solution set is the set of all real numbers
To solve a quadratic inequality, move all expressions to the left first
The boundary points of the solution set will be the points at which:
; that is, ; or,
; that is,
These values will be included in the solution set, since equality is allowed by the inequality symbol.
Test the intervals
by choosing a value in each interval and testing the truth of the inequality.
: test
False - exclude this interval
: test
True - include this interval
: test
False - exclude this interval
is the solution set.
Example Question #31 : Solving Inequalities
Solve for :
can be rewritten as the inequality
(note the change in direction of the inequality symbols)
This is the set .
Example Question #31 : Solving Inequalities
Solve the following inequality:
Like any other equation, we solve the inequality by first grouping like terms. Grouping the terms on the left side of the equation and the constants on the right side of the equation, we have: