Solving Linear Inequalities Involving Fractions

Need some practice without fractions first?

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

Example

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

Solution: Write a nice, clean list of equivalent sentences.

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

Concept Practice

Solve the given inequality. Write the result in the most conventional way.

For more advanced students, a graph is available. For example, the inequality $\,-\frac{2}{3}x + 6\le 1\,$ is optionally accompanied by the graph of $\,y = -\frac{2}{3}x + 6\,$ (the left side of the inequality, dashed green) and the graph of $\,y = 1\,$ (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 to toggle the graph.