Inequalities in Two Variables and Mixed Practice

From a point to a region#

An equa­tion like a straight line is sat­is­fied by points on a line. An inequal­ity in two vari­ables, such as $2x + 3y \le 12$, is sat­is­fied by a whole region of the plane: every­thing on one side of a line. Such regions are how we pic­ture real lim­its, like a fixed bud­get, as you will see in an exam­ple. We will graph such inequal­i­ties, use a test point to decide which side to shade, and find the region com­mon to a sys­tem of inequal­i­ties. A mixed prac­tice set on the whole chap­ter rounds things off.

One inequal­ity in two vari­ables#

The line $ax + by = c$ cuts the plane into two half-planes. Every point on one side sat­is­fies $ax + by < c$, and every point on the other side sat­is­fies $ax + by > c$. There is no mix­ing.

Method. First draw the line. Make it dashed for $<$ or $>$, since the line itself is not included, and solid for $\le$ or $\ge$, since it is. Then test any point not on the line; $(0, 0)$ is usu­ally eas­i­est. If it sat­is­fies the inequal­ity, shade its side. If not, shade the other side. Because of the no-mix­ing fact above, one test point is enough.

Sys­tems#

With sev­eral inequal­i­ties, the solu­tion is the region that every one of them shades. A prac­ti­cal tip: shade each one lightly, or bet­ter, mark the side each one rules out, so the com­mon region stands out clearly at the end.

Mixed prac­tice#

1. Solve for real $x$: $\displaystyle \frac{3x - 4}{2} \ge \frac{x + 1}{4} - 1$.
2. Solve: $\displaystyle 7 \le \frac{3x + 11}{2} \le 11$.
3. Solve the sys­tem $2(x - 1) < x + 5$, $3(x + 2) > 2 - x$.
4. Graph: $3x - 2y \ge 6$; $x + y < 4$; $y \le 3$.
5. Graph the sys­tem $2x + y \ge 4$, $x + y \le 5$, $x \ge 0$, $y \ge 0$ and list its cor­ner points.
6. Graph the sys­tem $x - 2y \le 3$, $3x + 4y \ge 12$, $x \ge 0$, $y \ge 1$.
7. Show that $x + y \ge 7$ and $2x + 2y \le 10$ have no com­mon solu­tion.
8. A com­pany makes chairs ($4$ h of labour each) and tables ($6$ h each), with at most $240$ labour hours. Write the inequal­ity and graph it for $x, y \ge 0$.
9. IQ $\displaystyle = \tfrac{MA}{CA} \times 100$. For $12$-year-olds the IQ ranges over $80 \le IQ \le 140$. Find the range of men­tal age $MA$.
10. Find all real $x$ sat­is­fy­ing both $\lvert x - 3 \rvert < 4$ and $x \ge 2$.

Answers#