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: [beautiful math coming... please be patient] $-6 - 3x \ge 4$
Solution:
[beautiful math coming... please be patient] $-6 - 3x \ge 4$ (original sentence)
[beautiful math coming... please be patient] $-3x \ge 10$ (add $\,6\,$ to both sides)
[beautiful math coming... please be patient] $x \le -\frac{10}{3}$ (divide both sides by $\,-3\,$; change the direction of the inequality symbol)
Solve: [beautiful math coming... please be patient] $3 - 2x \le 5x + 1$
Solution:
[beautiful math coming... please be patient] $3 - 2x \le 5x + 1$ (original sentence)
[beautiful math coming... please be patient] $3 - 7x \le 1$ (subtract $\,5x\,$ from both sides)
[beautiful math coming... please be patient] $-7x \le -2$ (subtract $\,3\,$ from both sides)
[beautiful math coming... please be patient] $x \ge \frac{2}{7}$ (divide both sides by $\,-7\,$; change the direction of the inequality symbol)
Solve: [beautiful math coming... please be patient] $\displaystyle -\frac{2}{3}x + 6\le 1$
Solution:
[beautiful math coming... please be patient] $\displaystyle -\frac{2}{3}x + 6\le 1$ (original sentence)
[beautiful math coming... please be patient] $-2x + 18\le 3$ (clear fractions; multiply both sides by $\,3\,$)
[beautiful math coming... please be patient] $-2x \le -15$ (subtract $\,18\,$ from both sides)
[beautiful math coming... please be patient] $\displaystyle x \ge \frac{15}{2}$ (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.

Solve: