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Solve linear inequalities in one variable

A linear inequality in one variable describes all values that make a linear comparison true, with solutions found by applying equivalent operations to isolate the variable. The solution is expressed as an interval, set notation, or a shaded number line; when multiplying or dividing by a negative number, the inequality symbol reverses, and compound inequalities represent intersections or unions of conditions. This scope excludes nonlinear, absolute-value, and multivariable inequalities.

Detailed Explanation: Solve linear inequalities in one variable

A linear inequality is solved much like a linear equation: isolate the variable using equivalent operations. The key difference is:

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

Example: Solve −3x+5≤17-3x+5\le 17.

  1. Subtract 55 from both sides:
−3x+5−5≤17−5-3x+5-5\le 17-5 −3x≤12-3x\le 12
  1. Divide both sides by −3-3. Since −3-3 is negative, reverse ≤\le to ≥\ge:
−3x−3≥12−3\frac{-3x}{-3}\ge \frac{12}{-3} x≥−4x\ge -4

Therefore, the solution is

x≥−4\boxed{x\ge -4}

In interval notation, this is [−4,∞)[-4,\infty). On a number line, use a closed circle at −4-4 because −4-4 is included, and shade to the right because the solutions are values greater than or equal to −4-4.

To check, use x=−4x=-4 in the original inequality:

−3(−4)+5=17-3(-4)+5=17

Since 17≤1717\le17 is true, the solution is consistent.

Learn by doing: Solve linear inequalities in one variable

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