Solving Linear Inequalities, All Mixed Up
This exercise mixes up problems from three earlier exercises:
- Solving Simple Linear Inequalities with Integer Coefficients
- Solving Linear Inequalities with Integer Coefficients
- Solving Linear Inequalities involving Fractions
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
$-6 - 3x \ge 4$ | 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 |
$3 - 2x \le 5x + 1$ | 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 |
$\displaystyle -\frac{2}{3}x + 6\le 1$ | 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.