Key takeaway: A two-step inequality is a statement with a comparison symbol that needs two inverse operations to isolate the variable, and its solution is a whole range of values, not one number.
Grade 77.EE.B.4bExpressions and Equations
Definition of Two-Step Inequality
A two-step inequality uses \(<\), \(\le\), \(>\) or \(\ge\) instead of an equal sign, and its variable is both multiplied and shifted by a constant. Solving a two-step inequality mirrors solving a two-step equation: clear the constant first, then the coefficient.
What you get at the end is a set of numbers, drawn as a ray. Exactly one new rule appears. Whenever the coefficient is negative, the final division turns the symbol around, so a “less than” becomes “greater than” and the other way round.
Formula and notation
A typical form is
\[ax + b \ge c\]
Solutions are graphed on a number line: a closed dot for \(\le\) or \(\ge\), an open dot for \(<\) or \(>\), with an arrow toward the values that work.
Key facts
- A negative multiplier or divisor flips the symbol; positive ones leave it alone.
- Adding or subtracting any number never changes the direction of the symbol.
- Closed dots include the endpoint, while open dots leave it out.
- Pick one value from the shaded ray and plug it in to confirm the answer.
Two-Step Inequality example
This two-step inequality models a goal. Maya has \$45 saved and earns \$6 each week walking dogs. She wants at least \$93 for a new helmet. Solve \(45 + 6w \ge 93\) to find how many weeks she needs.
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Subtract 45 from both sides: \(6w \ge 48\).
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Divide both sides by 6 (positive, so the symbol stays): \(w \ge 8\).
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Check with \(w = 9\): \(45 + 54 = 99 \ge 93\). True.
Maya needs at least 8 weeks (\(w \ge 8\)).
Common mistake
Watch out
Solving \(-4x + 3 > 19\), Ines subtracts 3 and divides by \(-4\) to get \(x > -4\).
Why it is wrong
Dividing by a negative number reverses the symbol. From \(-4x > 16\) the correct step gives \(x < -4\). Test \(x = -5\): \(-4(-5) + 3 = 23 > 19\), true. The value \(x = 0\), allowed by her answer, gives 3, which is not greater than 19.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Expressions and Equations