Linear inequalities

A linear inequality resembles a linear equation, but instead of an equality sign we have an inequality sign.

This means that the solution is not a single specific number, but often an entire interval of values.

A simple linear inequality could be:

$$ \large 2x + 3 < 7 $$

Here we are looking for the values of $ \large x $ that make the left-hand side less than 7.

A linear inequality is solved almost in the same way as a linear equation: we isolate the unknown by performing the same operations on both sides.

$$ \large 2x + 3 < 7 \quad \Leftrightarrow $$

$$ \large 2x < 7 - 3 \quad \Leftrightarrow $$

$$ \large 2x < 4 \quad \Leftrightarrow $$

$$ \large x < 2 $$

Thus, the solution is all values of $ \large x $ that are less than 2.

An important rule for inequalities is that when you multiply or divide by a negative number, the inequality sign must be reversed.

Example:

$$ \large -3x > 9 $$

We divide by $ \large -3 $, but remember to reverse the sign:

$$ \large x < -3 $$

Example 1:

$$ \large 5x - 7 \geq 3 $$

$$ \large 5x \geq 10 \quad \Leftrightarrow \quad x \geq 2 $$

Example 2:

$$ \large 4 - 2x \leq 10 $$

$$ \large -2x \leq 6 $$

Dividing by $ \large -2 $, we must reverse the sign:

$$ \large x \geq -3 $$

You can always check an inequality by inserting one value from the solution and one value outside the solution.

If the solution was $ \large x < 2 $:

Try $ \large x = 0 $:

$ \large 2\cdot 0 + 3 = 3 < 7 \quad $. True.

Try $ \large x = 3 $:

$ \large 2\cdot 3 + 3 = 9 < 7 \quad $. False.

A linear inequality resembles a linear equation, but the solution is often an interval of values.
You solve by isolating the unknown step by step.
When multiplying or dividing by a negative number, the inequality sign must be reversed.
You can always check the result by inserting test values.