Inequalities

One-Step Linear Inequality

Solve a linear inequality with a positive coefficient by isolating the variable. The direction of the inequality is preserved at each step because we only add/subtract and divide by a positive number.

Example

Solve a linear inequality while preserving the relation symbol.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

2x+3<112x + 3 < 11
Start with 2x + 3 < 11; the < sign is unchanged.2x + 3 < 11+3 constant< unchangedsubtractsubtract 3 from both sidesdividedivide by positive 24number lineshade left for less than

Step 2 — Subtract constant

Subtract 3 from both sides: 11 - 3 = 8.

2x<11−3=82x < 11 - 3 = \hl{8}
Subtract 3; subtraction keeps the direction.2x + 3 < 11+3 constant< unchangedsubtract2x < 11 - 3 = 8keeps directiondividedivide by positive 24number lineshade left for less than

Step 3 — Divide

Divide by positive 2; keep the sign: 8 / 2 = 4.

x<8/2=4x < 8 / 2 = \hl{4}
Divide by positive 2; the direction stays <.2x + 3 < 11+3 constant< unchangedsubtract2x < 11 - 3 = 8keeps directiondividex < 8 / 2 = 4positive 24number lineshade left for less than

Step 4 — Interval

Write the interval ending at 4.

x∈(−∞,4)x\in(-\infty, \hl{4} )
Graph x < 4 as (-∞, 4).2x + 3 < 11+3 constant< unchangedsubtract2x < 11 - 3 = 8keeps directiondividex < 8 / 2 = 4positive 24open circlex < 4 (-∞, 4)
solve-linear-inequality To solve ax + b < d with a > 0: subtract b from both sides (direction unchanged), then divide by a (direction unchanged). Express the solution set as an interval.