Linear Inequalities

A function of the form $f(x)=ax+b$ ($a,b\in R$ constants) is a linear function -- its graph is a straight line, with $a$ the slope and $b$ the $y$-intercept; if $a\ne0$, its $x$-intercept is $x=-\dfrac ba$ (solving $f(x)=0$).

Many real situations, however, are naturally expressed as linear inequalities rather than equations -- e.g. 'a tower is not taller than fifty feet' becomes $x\le50$ where $x$ is the tower's height.

Solving. Linear inequalities are solved exactly like linear equations, EXCEPT that multiplying or dividing by a negative number flips the inequality's direction (§2.2.4). A numeric example: an electricity bill of Rs.110 basic charge plus Rs.4/unit stays below Rs.250 exactly when $110+4x<250$, i.e. $4x<140$, i.e. (with $x\ge0$ since usage cannot be negative) $0\le x<35$ units.

Graphically, comparing two linear functions $f(x)=3x-5$ and $g(x)=x+1$: solving $f(x)\le g(x)$, i.e. $3x-5\le x+1\Rightarrow2x\le6\Rightarrow x\le3$, corresponds exactly to finding every $x$ where the graph of $f$ lies on or below the graph of $g$. …