Section 11.4: Solving Two-Step Inequalities

Property

To solve an equation with two or more operations, we must isolate the variable on one side of the equation. We undo the operations in reverse order. Typically, we undo addition or subtraction first, before undoing multiplication or division.

Examples

• To solve $4x + 5 = 29$, first subtract 5 from both sides to get $4x = 24$. Then, divide both sides by 4 to find $x = 6$.
• To solve $\frac{y}{3} - 2 = 7$, first add 2 to both sides to get $\frac{y}{3} = 9$. Then, multiply both sides by 3 to find $y = 27$.
• To solve $18 = 6 + 2z$, first subtract 6 from both sides to get $12 = 2z$. Then, divide both sides by 2 to find $z = 6$.

Explanation

Think of it as reversing your morning routine. To get back to the start, you undo the last thing you did first. In equations, this means handling addition or subtraction before dealing with multiplication or division to isolate the variable.

Property

A two-step linear inequality has the form $ax + b < c$, $ax + b \leq c$, $ax + b > c$, or $ax + b \geq c$, where $a \neq 0$. To solve a two-step inequality, use the same steps as solving a two-step equation, but reverse the inequality sign when multiplying or dividing both sides by a negative number.

Examples