Skill: Solve linear inequalities in one variable

A linear inequality is solved much like a linear equation: isolate the variable using equivalent operations. The key difference is:

When you multiply or divide both sides by a negative number, reverse the inequality symbol.

Example: Solve $-3x+5\le 17$.

1. Subtract $5$ from both sides:

$$-3x+5-5\le 17-5$$ $$-3x\le 12$$

2. Divide both sides by $-3$. Since $-3$ is negative, reverse $\le$ to $\ge$:

$$\frac{-3x}{-3}\ge \frac{12}{-3}$$ $$x\ge -4$$

Therefore, the solution is

$$\boxed{x\ge -4}$$

In interval notation, this is $[-4,\infty)$. On a number line, use a closed circle at $-4$ because $-4$ is included, and shade to the right because the solutions are values greater than or equal to $-4$.

To check, use $x=-4$ in the original inequality:

$$-3(-4)+5=17$$

Since $17\le17$ is true, the solution is consistent.