# Solving Linear Inequalities, All Mixed Up

This exercise mixes up problems from three earlier exercises:

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.

## 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

## 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 $\,3 - 2x \le 5x + 1\,$ 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 to toggle the graph.

Solve: