Drawing the answer as a piece of the number line, hollow circle or solid

What to assume they know

• Why testing values one by one is not a method, and what replaces it — solution set, and why a real-number answer cannot be listed
• The one rule that differs from equation-solving, and the reason it differs — the two rules, and the turn of the symbol under a negative multiplier
• Sorting inequalities: numerical or literal, strict or slack, linear or not — the strict and slack distinction, and double inequalities
• Plotting numbers on a number line, including negatives and simple fractions
• Union and intersection, and what it means for two sets to miss each other entirely — the intersection of two collections as the things lying in both
• The number systems as a chain of containments, and intervals as pieces of R — intervals as pieces of the real line, and the bracket notation this brief uses throughout

What they should be able to do

• Solve a linear inequality and draw its answer on a number line with the correct endpoint marking
• State what the hollow and filled circles mean, and connect each to one relation symbol
• Explain why the endpoint needs a mark of its own rather than being shown by where the shading stops
• Translate between a drawn answer and its bracket form
• Draw a system of two constraints as two lines and read the answer off the overlap
• Produce an answer that is closed at one end and open at the other, and say which constraint produced each end
• Recognise that an overlap can be unbounded rather than a finite segment
• State what "graphical representation" does and does not mean in this chapter

Where it usually goes wrong

• "The circle marks where the answer begins, so its style is decoration." The style is the only thing in the drawing that separates < from ≤ — but do not reach for Fig 5.1 and Fig 5.2 to show it. Those two differ in three ways at once: the mark sits at 3 in one and at 1 in the other, the heavy stroke runs left in one and right in the other, and only then does the mark style differ. The pair that isolates the style is inside this chapter's own exercise: item 18 of Exercise 5.1 resolves to x ≥ −1 and item 19 to x > −1 — same endpoint, same direction, nothing left to separate them but the mark.
• "There must be a largest solution to x < 3, just very close to 3." No such number exists; the midpoint construction produces a larger one every time. That is precisely why the endpoint has to be marked rather than shown.
• "The arrowhead means the line stops there." It means the shading carries on past the edge of the drawing.
• "Round and square brackets are two styles for the same thing." They are the hollow and the filled circle written down.
• "For a system, shade one line and be done." Each constraint gets its own line; the answer is where they overlap. Fig 5.3 draws three lines for two constraints for exactly this reason.
• "An overlap is always a finite segment." Miscellaneous Exercise item 9's overlap is a ray with no right-hand end.
• "Both ends of an answer must be the same kind." In Fig 5.3 one is filled and one is hollow, and each traces back to a different constraint's symbol.
• "Graphical representation means drawing it in the plane." Not in this chapter. Every drawing here is on a single line.

Questions to check understanding

• Solve a linear inequality for real x and draw the answer, with the endpoint marked correctly
• Write a drawn answer in bracket form, and the reverse
• Solve a system of two linear inequalities in one letter and represent it on a number line
• Say which of two given drawings matches a stated inequality, when they differ only in the endpoint mark
• Given an answer closed at one end and open at the other, say which constraint produced each end
• Decide whether a stated value lies in the answer, in a case where it is exactly an endpoint

Examples worth working on the board

Items marked verified are worked out here from the chapter's stated data. The book's separate answers file was not read, checked or extracted at any point.

• Example 5 and Fig 5.1 (§5.3, p. 93). The statement is 7x + 3 < 5x + 9, which the chapter reduces to 2x < 6 and then x < 3. Fig 5.1, read off the page image: a horizontal line with ticks labelled −4 through 6, a hollow circle sitting on 3, and a heavy stroke running left from that circle and ending in an arrowhead at the left edge.
• Example 6 and Fig 5.2 (§5.3, pp. 93–94). The statement is (3x − 4)/2 ≥ (x + 1)/4 − 1, which the chapter reduces to 2(3x − 4) ≥ x − 3, then 6x − 8 ≥ x − 3, then 5x ≥ 5 and x ≥ 1. Fig 5.2, read off the page image: ticks again labelled −4 through 6, a filled dot on 1, and the heavy stroke running right with an arrowhead. Two drawings, the same line, two different marks — the contrast is the lesson.
• Verified, section 3, and this is the argument the whole topic rests on: pick any real number r below 3. The midpoint (r + 3)/2 lies strictly between r and 3, so it is also a solution and it is larger than r. Repeat and the candidates crowd towards 3 without ever reaching a last one. A drawing therefore cannot indicate exclusion by stopping the shading somewhere; the only place left to put the information is a mark on the endpoint itself.
• The bracket forms already in use (§5.3, pp. 92–93). Example 2's real-number answer is written as everything below 2 with a round bracket at 2; Example 3's answer runs upward from −2 with a round bracket; Example 4's runs upward from 8 with a square bracket. Verified correspondence to state plainly: a round bracket is the hollow circle written down, a square bracket is the filled one.
• Miscellaneous Example 11 and Fig 5.3 (pp. 96–97). Two constraints on the same letter: 3x − 7 < 5 + x, which the chapter reduces to x < 6; and 11 − 5x ≤ 1, which reduces to x ≥ 2. Fig 5.3, read off the page image: three horizontal lines stacked. The topmost carries a filled dot at 2 with its stroke running right; the middle carries a hollow circle at 6 with its stroke running left; the lowest is the full axis, ticks labelled −1 through 9, with a filled dot at 2, a hollow circle at 6, and the stretch between them drawn heavy. The chapter's answer is that x lies from 2 up to 6, with 2 in and 6 out.
• Verified, and this is section 10: the two ends get their marks for two different reasons, each traceable to a single symbol. At x = 2 the first constraint gives −1 against 7, comfortably true, and the second gives 1 against 1 — equality, which the slack symbol admits, so 2 is in and it is ≤ that put it there. At x = 6 the first constraint gives 11 against 11, and the strict symbol refuses equality, so 6 is out. One answer, two endpoint rules, both readable off the original constraints.
• Exercise 5.1 items 17 to 20 (p. 95) ask for the drawing as well as the answer. Item 17: 3x − 2 < 2x + 1. Item 18: 5x − 3 ≥ 3x − 5. Item 19: 3(1 − x) < 2(x + 4). Item 20: x/2 ≥ (5x − 2)/3 − (7x − 3)/5. Text-layer trap, and it changes the picture: the extraction of item 18 loses the slack half of its relation and shows a strict one. The printed relation is slack. A strict relation there would give a hollow circle where the page requires a filled one.
• Verified, item 17: it reduces to x < 3, which is Fig 5.1 again — a deliberate echo of the worked example.
• Verified, items 18 and 19, which are the best pair in the exercise: item 18 reduces to 2x ≥ −2 and so to x ≥ −1, a filled dot at −1 shading right; item 19 reduces to 3 − 3x < 2x + 8, then −5 < 5x and so x > −1, a hollow circle at −1 shading right. Same endpoint, same direction, different mark. Put them on adjacent lines.
• Verified, item 20: multiplying through by 30 gives 15x ≥ 10(5x − 2) − 6(7x − 3), which is 50x − 20 − 42x + 18, that is 8x − 2. So 7x ≥ −2 and x ≥ −2/7 — a filled mark that does not sit on a tick. Worth showing: endpoints are not obliged to be integers.
• Miscellaneous Exercise items 7 to 10 (p. 98) each ask for a system drawn on the number line. Item 7: 5x + 1 > −24 together with 5x − 1 < 24. Item 8: 2(x − 1) < x + 5 together with 3(x + 2) > 2 − x. Item 9: 3x − 7 > 2(x − 6) together with 6 − x > 11 − 2x. Item 10: 5(2x − 7) − 3(2x + 3) ≤ 0 together with 2x + 19 ≤ 6x + 47.
• Verified, and these four. Item 7 gives x > −5 and x < 5, so the answer runs from −5 to 5 with both ends hollow. Item 8 gives x < 7 and x > −1, so it runs from −1 to 7, again both ends hollow. Item 9 gives x > −5 and x > 5, and the second constraint swallows the first, so the answer is everything above 5 — an overlap that is a ray, not a segment. Item 10 gives x ≤ 11 and x ≥ −7, so it runs from −7 to 11 with both ends filled.
• The Summary (p. 99) states the convention in two bullets, one for the strict pair and one for the slack pair, and it is where the words for the two marks are actually printed. Read on the page image, the slack bullet ends by naming the letter standing for the variable in the place where the endpoint's own name belongs. Treat it as a printing slip; the two figures on pp. 93 and 94 show what is meant.

Figures to have open

• Fig 5.1 redrawn as a schematic: ticks −4 through 6, hollow mark at 3, heavy stroke running left to an arrowhead. This is the chapter's own figure (p. 93); redraw it rather than reproducing the printed art.
• Fig 5.2 redrawn the same way: ticks −4 through 6, filled mark at 1, heavy stroke running right to an arrowhead (p. 94).
• Fig 5.3 redrawn as three stacked lines: filled mark at 2 shading right, hollow mark at 6 shading left, and beneath them the axis with ticks −1 through 9 carrying both marks and the heavy stretch between (p. 97). The stacking is the whole point and must survive the redraw.
• A "no last member" panel for section 3, with a candidate and its midpoint produced twice. An added figure; the chapter argues nothing here.
• A paired line for Exercise 5.1 items 18 and 19, both endpoints at −1, both shading right, one filled and one hollow. An added construction from the chapter's exercise data.
• Nothing in this topic requires a coordinate plane. No drawing in pp. 89–99 has two axes — checked on all eleven page images.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 5 Linear Inequalities, §5.3, p. 93, for Example 5 and Fig 5.1; pp. 93–94 for Example 6 and Fig 5.2.
• Miscellaneous Examples, pp. 96–97, for Example 11 and Fig 5.3.
• §5.3, pp. 92–93, for the bracket forms attached to the answers of Examples 2, 3 and 4.
• Exercise 5.1 items 17 to 20, p. 95; Miscellaneous Exercise on Chapter 5 items 7 to 10, p. 98.
• Summary, p. 99, for the two bullets stating the endpoint convention and for the printed words naming the two marks.