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All courses Math Foundations · MF.EQ.11 38 of 60
Solve multi-step inequalities, including reversing the inequality when appropriate.

Combine like terms before solving an inequality

Problem
Combine like terms on the left side, then solve the inequality: 3x + 5 + x > 17
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Hint

Start by combining the like terms on the left: the two x-terms go together, but the 5 stays separate.

Only like terms combine. After simplifying, solve the one-step inequality by undoing +5, then dividing by a positive coefficient, so the > direction stays the same.

Solution walkthrough

01

Combine the variable terms on the left

\[3x~+~5~+~x~>~17~\;\rightarrow\;~4x~+~5~>~17\]

The like terms are the variable terms \(3x\) and \(x\). Since \(3x~+~x~=~4x\), the inequality simplifies to \(4x~+~5~>~17\).

02

Subtract 5 from both sides

\[4x~+~5~>~17~\;\rightarrow\;~4x~>~12\]

Subtracting the same number from both sides keeps the inequality equivalent and moves the constant away from the variable term.

03

Divide by the coefficient of x

\[4x~>~12~\;\rightarrow\;~x~>~3\]

Dividing both sides by \(4\) isolates \(x\). Because \(4\) is positive, the inequality direction stays \(>\).

04

State the solved inequality

\[x~>~3\]

Any number greater than \(3\) makes \(3x~+~5~+~x~>~17\) true, so the solution is \(x~>~3\).

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Another way

  1. Check the boundary and a nearby value: at \(x~=~3\), \(3(3)~+~5~+~3~=~17\), which is not greater than \(17\), so \(3\) is not included. At \(x~=~4\), \(3(4)~+~5~+~4~=~21\), and \(21~>~17\), so values greater than \(3\) work.

!

Common mistake

A common mistake is combining \(3x~+~5~+~x\) into \(9x\). The \(5\) is a constant, not an \(x\) term, so only \(3x\) and \(x\) combine. The correct simplification is \(4x~+~5\).