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Basic Linear Inequalities (single variable)
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You have already worked with inequality statements. Let's refresh those skills.

Inequality Notations: (see other notation forms at Notations for Solutions)
a > b ;    a is strictly greater than b
a greatereqal b ;   a is greater than or equal to b
a < b ;    a is striclty less than b
a lessequal 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.

 

statement
If you can solve a linear equation, you can solve a linear inequality. The process is the same, with one 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.
refreshgraph1
-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.

 

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".

circleg1                circleg2

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ex1

Graph the solution set of:   -3 < x < 4
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.

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

circleg3


ex1

Graph the solution set of:   x < -3 or x greaterE1
The solution set for this problem will be the full graph of both inequalities, since the two inequalities do not overlap.

Notice that there is one open circle (for -3) and one closed circle (for 1).

circlegraph10a



ex1

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).

iqmath7

circleg3



ex2

Solve and graph the solution set of:   -5x greaterequala 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).

iqmath8

circlegraph8a



ex3

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).

iqmath1

circleg3



ex3

Solve and graph the solution set of:   9 - 2x lessequal 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).

iqmath4

circleg4



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