audio read-through 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: