Two-Step Inequality
In short: Two-step inequality refers to a statement like 4x – 9 > 23 where the unknown is multiplied and shifted, and two undoing moves reveal a whole range of values that satisfy it.
7th GradeAlgebra7.EE.B.4b
What is Two-Step Inequality?
A two-step inequality such as \(ax + b \le c\) fits situations with a budget or a limit, where you need every answer that works and not just one exact value, so its solution is a whole range of numbers.
Work it the way you would an equation: cancel the added or subtracted piece first, then the multiplier. Only one habit changes. Whenever the multiplier is negative, the arrow of the comparison has to turn around.
How to spot it
- A story contains the words at most, at least, no more than, fewer than or more than.
- A budget or weight cap is tied to a flat fee and a charge for each item.
- A comparison sign such as \(\le\) or \(\ge\) stands where an equals sign would sit.
- The answer is a set of values to graph on a number line, not one number.
The method
- Undo the number added to or subtracted from the variable term on both sides.
- Divide both sides by the variable’s coefficient; if that number is negative, turn the sign around.
- Write the result with the variable on the left, such as \(x > 8\).
- Graph it: hollow dot when the endpoint is excluded, solid dot when it is included, then test one value.
Two-Step Inequality: solved example
Problem
Tomás has \(\$60\) for a fair. Entry costs \(\$12\) and each ride ticket costs \(\$4.50\). What is the greatest number of rides he can buy?
Solution
Write \(12 + 4.5n \le 60\). Subtract 12: \(4.5n \le 48\). Divide by 4.5: \(n \le 10.\overline{6}\).
Rides come in whole numbers, so the largest allowed value is 10 rides. Check: \(12 + 45 = 57 \le 60\), while 11 rides cost \(61.50\).
Try it
Question 1
Solve \(-3x + 7 \ge 25\) and describe the direction of the sign.
Show the answer
Subtract 7: \(-3x \ge 18\). Dividing by \(-3\) flips the sign: \(x \le -6\).
Question 2
Solve \(\frac{n}{5} + 4 < 9\).
Show the answer
Subtract 4: \(\frac{n}{5} < 5\). Multiply by 5: \(n < 25\).
Two-Step Inequality vs. Two-Step Equation
A two-step equation has one answer, and you can check it by plugging in that number. A two-step inequality has infinitely many answers, graphed as a ray, and the sign may flip when you divide by a negative. A single test value inside the solution set is the quick check for the inequality.
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Whole dictionary: A–Z · 7th Grade · Algebra