Not all statements in mathematics involve dealing with equal quantities.
Sometimes, we may only know that something is "greater than" a certain value,
or "less than or equal to" another value.
These situations are referred to as inequalities (because they are not simply "equal").

 Inequality Notations: (see other notation forms at Notations for Solutions) a > b ;    a is strictly greater than b a b ;   a is greater than or equal to b a < b ;    a is striclty less than b a b ;    a is less than or equal to b a ≠ b ;     a is not equal to b Hint: The "open" (larger) part of the inequality symbol always faces the larger quantity.

 If you can solve a linear equation, you can solve a linear inequality. The process is the same, with one important exception ...

 ... when you multiply (or divide) an inequality by a negative value, you must change the direction of the inequality.

Let's see why this "exception" is actually needed.

 We know that 3 is less than 7. Now, lets multiply both sides by -1. Examine the results (the products). ... written 3 < 7. ... written (-1)(3) ? (-1)(7) ... written -3 ? -7 On a number line, -3 is to the right of -7, making -3 greater than -7. -3 > -7 We have to reverse the direction of the inequality, when we multiply by a negative value, in order to maintain a "true" statement.

 When graphing a linear inequality on a number line, use an open circle for "less than" or "greater than", and a closed circle for "less than or equal to" or "greater than or equal to".

To CHECK an inequaltiy, it is not possible to test every value.
So check a value in each
shaded region to see if it is TRUE.
Then check a value in each non-shaded region to see if it is FALSE.

 Graph the solution set of:    x -3 The solution set for this problem will be all values that are graphed to the right of -3, and including -3. Graph using closed circle for -3 since x = -3 is among the solutions. CHECK: A number in the shaded region = TRUE. A number in the non-shaded area = FALSE. Pick 0:   0 > -3 TRUE Pixk -4:   -4 > -3 FALSE

 Graph the solution set of:   -3 < x < 4 AND
The solution set for this problem will be all values that satisfy both -3 < x and x < 4. Look for where the two inequalities overlap.
The solution will be all x's that are greater than -3 and at the time are less than +4.

Graph using open circles for -3 and 4 (since x can not equal -3 nor 4), and a bar to show the overlapping section.

This type of inequality may be referred to as a "sandwich" since the solutions are sandwiched between -3 and +4.

CHECK: pick 1: -3 < 1 < 4 TRUE
pick -4: -3 < -4 < 4 FALSE
pick 5: -3 < 5 < 4 FALSE

 Graph the solution set of:   x < -3 or x 1 OR
The solution set for this problem will be the full graph of both inequalities, since the two inequalities do not overlap.

in x < -3 or in x > 1.

Notice that there is one open circle
(for -3) and one closed circle (for 1).
CHECK: Remember "OR" is true when either (or both) section is true, and false when both are false.
pick -4: -4 < -3 OR -4 > 1 This is TRUE since -4 < -3 is true.
pick 3: 3 < -3 OR 3 > 1 This is TRUE since 3 > 1 is true.
pick -1: -1 < -3 OR -1 > 1 This is FALSE since both sections are false.

 Solve and graph the solution set of:   4x < 24 Proceed as you would when solving a linear equation: Divide both sides by 4. Note: The direction of the inequality stays the same since we did NOT divide by a negative value. Graph using an open circle for 6 (since x can not equal 6) and an arrow to the left (since our symbol is less than). CHECK: pick 1: 1 < 6 TRUE pick 7: 7 < 6 FALSE

 Solve and graph the solution set of:   -5x 25 Divide both sides by -5. Note: The direction of the inequality was reversed since we divided by a negative value. Graph using a closed circle for -5 (since x can equal -5) and an arrow to the left (since our final symbol is less than or equal to). CHECK: pick -7: -7 < -5 TRUE pick 0: 0 < -5 FALSE

 Solve and graph the solution set of:   3x + 4 > 13 Proceed as you would when solving a linear equation: Subtract 4 from both sides. Divide both sides by 3. Note: The direction of the inequality stays the same since we did NOT divide by a negative value. Graph using an open circle for 3 (since x can not equal 3) and an arrow to the right (since our symbol is greater than). CHECK: pick 4: 4 > 3 TRUE pick 0: 0 > 3 FALSE

 Solve and graph the solution set of:   9 - 2x 3 Subtract 9 from both sides. Divide both sides by -2. Note: The direction of the inequality was reversed since we divided by a negative value. Graph using a closed circle for 3 (since x can equal 3) and an arrow to the right (since our symbol is greater than or equal to). CHECK: pick 4: 4 > 3 TRUE pick 0: 0 > 3 FALSE

 Solve and graph the solution set of:   5 - 2x 13 + 2x Add 2x to both sides. Subtract 13 from both sides. Divide both sides by 4. Note: There was no multiplication or division by a negative value, so the inequality symbol did not get reversed. Graph using a closed circle for -2 (since x can equal -2). CHECK: pick 0: 0 > -2 TRUE pick -4: -4 > -2 FALSE

 Solve and graph the solution set of:   4x + 10 < 3(2x + 4) Distribute across parentheses. Subtract 4x from both sides. Add -12 to both sides. Divide both sides by 2. Note: There was no multiplication or division by a negative value, so the inequality symbol did not get reversed. Graph using a closed circle for -2 (since x can equal -2). CHECK: pick 0: 0 > -1 TRUE pick -3: -3 > -1 FALSE

 Yes, there is a way to determine solutions for inequalities on your graphing calculator. Click the calculator at the right to see how to use the calculator with single variable inequalities.