How this calculation works
Collect variable terms and constants as for a linear equation. A positive divisor preserves the comparison direction, while a negative divisor reverses it. Strict relations exclude the boundary and inclusive relations include it. A zero collected coefficient leaves a constant true-or-false statement.
(a−c)x relation d−b; dividing by a negative coefficient reverses the relationA worked example
For 2x+3 < x+7, collect terms to obtain x < 4. The boundary 4 is excluded. For −2x+3 < 7, divide −2x < 4 by −2 and reverse the sign: x > −2.
Good to know
One real linear inequality only, with entered decimal coefficients and 40 significant-digit working precision. No compound chains, absolute-value cases, quadratic inequalities or variable denominators. Rounded screen values are not exact endpoint representations for recurring decimal boundaries.
Reference: OpenStax: solving linear inequalities (opens in a new tab) ↗
A solution set can contain many values
An inequality asks which inputs make one expression smaller or larger than another. Its answer is generally a region of the number line rather than a single value. The boundary is useful, but knowing that boundary alone does not tell you which side is allowed or whether the boundary belongs to the solution set.
The linear equation solver finds the equality boundary for a nondegenerate pair of linear expressions. The inequality version keeps the strict or inclusive comparison and checks the sign of the coefficient before dividing. Those two extra details determine the actual set of allowed values.
A negative divisor reverses the comparison
For −2x+3<7, subtract three to obtain −2x<4. Dividing both sides by −2 gives x>−2. A quick check at x=0 gives 3<7, which is true. Keeping the original less-than sign after dividing by a negative number would incorrectly exclude that valid test point.
Use the relation selector deliberately. With x<4, the value four is excluded; with x≤4 it is included. The table checks a value on each side and treats boundary inclusion algebraically. A recurring-decimal boundary printed to a limited number of digits is a rounded display, not an exact replacement for the underlying ratio.
An absent variable does not produce an infinite boundary
If a and c are equal, the variable terms cancel. The remaining comparison is either always true or always false. For 2x+3<2x+7, every real number works because 3<7. For 2x+3>2x+7, no real number works. There is no meaningful division by zero to perform.
This form covers one linear inequality, not a chained interval such as 1<x<5 or an absolute-value inequality. Coefficients represent the expressions ax+b and cx+d only. If a word problem has several simultaneous limits, solving each supported inequality separately is only one step toward checking their intersection.
A few questions, answered
Why did less than change to greater than?
Dividing both sides by a negative collected coefficient reverses their order.
Does an inclusive symbol include the boundary?
Yes. The nondegenerate equality boundary is included for ≤ or ≥ and excluded for < or >.
Does every real number include negative numbers?
Yes. It includes all real x, including negative values, zero and positive values.