# 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.