Linear Inequalities

Reverse the comparison only for negative multiplication or division

For $7-3x>16$, subtract 7 to get $-3x>9$. Dividing by $-3$ reverses the sign: $x<-3$. Test $x=-4$: the original left side is 19, which is greater than 16.

Moving a term by adding or subtracting does not itself reverse an inequality. The reversal comes from multiplying or dividing both sides by a negative quantity.

Translate limits into inclusive or strict boundaries

A budget of at most 50 dollars gives a $\le$ constraint. A height greater than 120 centimeters gives a $>$ constraint. Equality is allowed in the first case and excluded in the second.

If notebooks cost 4 dollars each and shipping is 6 dollars, a 30-dollar budget gives $4n+6\le 30$, hence $n\le 6$. Since $n$ counts notebooks, use nonnegative whole numbers, not every real value below 6.

Test points in two-variable inequalities

For $y<2x+1$, draw the boundary $y=2x+1$ as a dashed line because equality is excluded. The point $(0,0)$ works since $0<1$, so the side containing that point is shaded.

A system requires the overlap. A point satisfying only one inequality is not feasible. Substitution is a direct check when the answer choices are ordered pairs.

Check maximum and minimum requests

If $2p+3q\le 18$ with nonnegative integers $p$ and $q$, the maximum $q$ occurs at $p=0$, giving $q=6$. Adding an extra condition such as $p\ge 2$ changes the bound to $q\le \frac{14}{3}$, so the largest integer is 4.

A numerical boundary is not always an allowed answer. Apply the integer restriction and strictness after the algebra, then test the proposed endpoint.

Check your understanding

Sample linear inequalities questions

Practice this skill

1. Translate each phrase into a comparison symbol before solving.
2. Test a boundary and one interior value in the original inequality.
3. For two-variable systems, check each constraint separately.

Print the linear inequalities worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.