Skill: Solve one-variable linear inequalities

To solve a one-variable linear inequality, isolate the variable just as you would when solving an equation. The key difference is:

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

Example

Solve:

$$5-2(x+1)<x-4$$

Step 1: Distribute and simplify.

$$5-2x-2<x-4$$ $$3-2x<x-4$$

Step 2: Move the variable terms to one side.

Subtract $x$ from both sides:

$$3-3x<-4$$

Step 3: Move the constant term.

Subtract $3$ from both sides:

$$-3x<-7$$

Step 4: Divide by $-3$.

Because $-3$ is negative, reverse the inequality symbol:

$$x>\frac{7}{3}$$

Therefore, the solution is

$$\boxed{x>\frac{7}{3}}$$

On a number line, use an open circle at $\frac{7}{3}$ because the inequality is $>$, not $\geq$, and shade to the right. In interval notation, the solution is

$$\left(\frac{7}{3},\infty\right).$$