Examples of first-degree inequalities

Two examples show the basic mechanics: in the first the solution is immediate, in the second we must reverse the direction because we divide by a negative number. In both cases the solution is an interval that we represent on the real line.

Example — Simple first-degree inequality

Solve $3x-7 > 2x+1$. $3x-2x > 1+7 \implies x > 8.$ The solution is the interval $\boxed{x > 8}$, that is $(8,+\infty)$.

Representation on the line:

The open circle at $x=8$ indicates that $8$ is not part of the solution (the inequality is strict, $>$ and not $\ge$).

Example — Sign reversal

Solve $-2x+5 \ge 3$. $-2x \ge -2 \;\Longrightarrow\; x \le 1$ Dividing by $-2$ (a negative number) the sign $\ge$ becomes $\le$. Solution: $\boxed{x\le 1}$.

The filled circle at $x=1$ indicates that $1$ belongs to the solution ($\le$, not $<$).

Links

Topics: Inequalities
Concepts: Sign reversal · First-degree inequality · Interval
Skills: Interpreting a graph · Solving inequalities