A two-step inequality compares an expression that needs two operations to undo with a number, such as \(2x+3>11\). Solve it like a two-step equation: undo the addition or subtraction first, then the multiplication or division. If you multiply or divide both sides by a negative number, reverse the inequality sign.
Key facts:
- Standard 7.EE.B.4b: solve \(px+q>r\) and \(px+q< r\) and graph the solution set.
- Undo addition or subtraction first, then multiplication or division.
- Multiplying or dividing by a negative number reverses the inequality sign.
- Use an open circle for \(<\) and \(>\), and a closed circle for \(\le\) and \(\ge\).
- In a word problem, read the solution set in context, for example whole items only.
What is a two-step inequality?
A two-step inequality has a variable that is multiplied or divided and also has a number added or subtracted, like \(2x+3>11\). Unlike an equation, it has a whole solution set: every number that makes the statement true.
Inequality symbols: \(<\) less than, \(>\) greater than, \(\le\) less than or equal to, \(\ge\) greater than or equal to.
Graph: an open circle for \(<\) and \(>\), a closed circle for \(\le\) and \(\ge\), and a shaded ray toward the solutions.
How do you solve a two-step inequality?
Treat it like a two-step equation. Undo addition or subtraction first, then undo multiplication or division, doing the same thing to both sides.
- Add or subtract to get the variable term alone.
- Multiply or divide to get the variable alone.
- If you multiplied or divided by a negative number, reverse the symbol.
- Graph the solution and check one number.
Worked example. Solve \(3x-5< 10\). Add 5: \(3x< 15\). Divide by 3: \(x< 5\). Check \(x=4\): \(3(4)-5=7< 10\), which is true.
When do you reverse the inequality sign?
Whenever you multiply or divide both sides by a negative number. Negative numbers flip the order on the number line, so \(-5< 2\) becomes \(5>-2\) after multiplying by \(-1\). Adding or subtracting never changes the sign.
Example: \(-2x+1< 9\) gives \(-2x< 8\). Dividing by \(-2\) gives \(x>-4\), not \(x<-4\).
How do you use a two-step inequality in a real situation?
Define the variable, write the inequality, solve it, then decide what the answer means. Jamal has \(\$50\). A cap costs \(\$12\) and each shirt costs \(\$9\), so \(12+9s\le 50\) and \(s\le 4\frac{2}{9}\). Because he cannot buy part of a shirt, the most he can buy is 4 shirts.
| Words | Symbol | Circle |
|---|---|---|
| at least, no less than | \(\ge\) | closed |
| at most, no more than | \(\le\) | closed |
| more than, greater than | \(>\) | open |
| fewer than, less than | \(<\) | open |
Two-Step Inequalities: frequently asked questions
What is a two-step inequality?
It is an inequality such as \(4x-7\ge 9\) that needs two inverse operations to isolate the variable. Its answer is a set of numbers, not one number.
When do you flip the inequality sign?
Flip it only when you multiply or divide both sides by a negative number. Adding or subtracting a negative number does not change the sign.
How do you graph \(x\ge 4\)?
Draw a closed circle at 4 because 4 is included, then shade the number line to the right, toward larger numbers.
How do you check a solution?
Pick a number inside the shaded region and substitute it into the original inequality. It should make a true statement. A number outside the region should make it false.
Why does a word problem sometimes need a whole-number answer?
Items such as shirts or tickets cannot be split. After solving, choose the greatest or least whole number that fits the solution set and the situation.
See every chapter on the Grade 7 math page.