Solving Linear Inequalities, all mixed up

Remember that whenever you multiply or divide both sides of an inequality by a negative number,
then you must change the direction of the inequality symbol.

EXAMPLES:

Solve:

$-6 - 3x \ge 4$

Solution:

$-6 - 3x \ge 4$	(original sentence)
$-3x \ge 10$	(add $\,6\,$ to both sides)
$x \le -\frac{10}{3}$	(divide both sides by $\,-3\,$; change the direction of the inequality symbol)

Solve:

$3 - 2x \le 5x + 1$

Solution:

$3 - 2x \le 5x + 1$	(original sentence)
$3 - 7x \le 1$	(subtract $\,5x\,$ from both sides)
$-7x \le -2$	(subtract $\,3\,$ from both sides)
$x \ge \frac{2}{7}$	(divide both sides by $\,-7\,$; change the direction of the inequality symbol)

Solve:

$\displaystyle -\frac{2}{3}x + 6\le 1$

Solution:

$\displaystyle -\frac{2}{3}x + 6\le 1$	(original sentence)
$-2x + 18\le 3$	(clear fractions; multiply both sides by $\,3\,$)
$-2x \le -15$	(subtract $\,18\,$ from both sides)
$\displaystyle x \ge \frac{15}{2}$	(divide both sides by $\,-2\,$; change the direction of the inequality symbol)

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Simplifying Basic Absolute Value Expressions

For more advanced students, a graph is displayed.
For example, the inequality $3 - 2x \le 5x + 1$
is optionally accompanied by the graph of $\,y = 3 - 2x\,$ (the left side of the inequality, dashed green)
and the graph of $\,y = 5x + 1\,$ (the right side of the inequality, solid purple).
In this example, you are finding the values of $\,x\,$ where the green graph lies on or below the purple graph.
Click the “show/hide graph” button if you prefer not to see the graph.