Solve one-variable linear inequalities

Solving a one-variable linear inequality involves transforming an inequality such as $ax+b<c$ into an equivalent statement that identifies all values of the variable satisfying the constraint, including cases with rational coefficients, parentheses, and variables on both sides. The solution set is represented on a number line or in interval notation, with endpoint inclusion determined by the inequality symbol; multiplying or dividing by a negative reverses the inequality direction. Systems, absolute-value inequalities, and nonlinear inequalities are outside this scope.

Detailed Explanation: Solve one-variable linear inequalities

To solve a one-variable linear inequality, isolate the variable just as you would when solving an equation. The key difference is:

When you multiply or divide by a negative number, reverse the inequality symbol.

Example

Solve:

5-2(x+1)<x-4

Step 1: Distribute and simplify.

5-2x-2<x-4

3-2x<x-4

Step 2: Move the variable terms to one side.

Subtract $x$ from both sides:

3-3x<-4

Step 3: Move the constant term.

Subtract $3$ from both sides:

-3x<-7

Step 4: Divide by $-3$ .

Because $-3$ is negative, reverse the inequality symbol:

x>\frac{7}{3}

Therefore, the solution is

\boxed{x>\frac{7}{3}}

On a number line, use an open circle at $\frac{7}{3}$ because the inequality is $>$ , not $\geq$ , and shade to the right. In interval notation, the solution is

\left(\frac{7}{3},\infty\right).

Learn by doing: Solve one-variable linear inequalities

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Two-Step Inequality - Subtraction and Multiplication - Solve

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