Solving and Graphing Inequalities: Solve It, Then Draw It

The waterslide sign does not list qualifying riders — 48.5 inches, 49, 61. It names one cutoff, 48 inches, and the picture covers everybody else.

A solid dot on 48, arrow running right: this value and every bigger one, in one glance.

A linear inequality is a one-variable sentence using one of the four signs $<$, $\le$, $>$, $\ge$ — like $h \ge 48$: “at least 48 inches” means greater than or equal to, so $h \ge 48$. Test it: $50 \ge 48$ is true, while $47 \ge 48$ is false.

Solving and graphing inequalities is the craft of producing that picture from any starting inequality. Solving and graphing inequalities always ends on a number line — cutoff marked, working side shaded.

Every “solve each inequality and graph its solution” problem asks for the same four moves in order:

1. Tidy both sides. Add or subtract to gather variable terms on one side, numbers on the other — this never disturbs the sign.
2. Isolate the variable. Divide by its coefficient: positive leaves the sign alone, negative flips it.
3. Name the cutoff. The solution, say $x > 3$, states the boundary and the surviving side.
4. Draw it. Solid dot or open circle at the boundary, shading toward the values that work.

The flip is the only “unsafe” move; for example, $-2x > 4$ is divided by $-2$, the sign flips, and $x < -2$ appears.

Solving and graphing inequalities needs nothing fancier than these four moves. Solving and graphing inequalities just needs them in order, every time.

Three clues carry the whole answer.

• The endpoint mark. A parenthesis means excluded, a bracket included; on the line an open circle plays the parenthesis, a solid dot the bracket.
• The direction. Right means greater; left means less.
• The boundary number. The dot sits on the cutoff.

Interval notation agrees: $x > 3$ is $(3, \infty)$, $x \le 3$ is $(-\infty, 3]$.

Solving and graphing inequalities writes every answer twice — as symbols and as a drawing — and the two versions must agree.

Solving and graphing inequalities finishes at this table: pick the row, draw the row. Solving and graphing inequalities in reverse starts here too — a finished graph reads back through the same three clues.

Dividing by a negative rewires the sign. Proof: $4 > 2$; times $-1$ gives $-4 < -2$. The order flipped at zero — the flip solving and graphing inequalities inherits.

Durell's 1911 School Algebra states the same law in older ink: both signs open toward the greater quantity — the working fact behind solving and graphing inequalities. Wentworth's 1898 New School Algebra agrees: the symbol points at the smaller number.

So solving and graphing inequalities treats the flip as a checkpoint, not a trick: pause at every negative divide — that pause is where the grade is won or lost.

Problem. Solve the inequality and graph the solution.

Divide by 8 — positive, so the sign stays: $x < 6$.

Answer. $x < 6$ — open circle at 6, shading left. The circle is open because 6 itself fails: $8 \cdot 6 = 48$ is not less than 48. One divide is the lightest case of solving and graphing inequalities: one move, one drawing.

Check. Test point $x = 0$: $0 < 6$, true. Off the shading, $x = 9$: $9 < 6$, false.

Solving and graphing inequalities closes the same way every time — cutoff in the middle, arrow pointing at the survivors. Read the finished line back and solving and graphing inequalities has handed you both answers at once.

Problem. Solve the inequality and graph the solution.

Divide both sides by $-4$. The divisor is negative, so flip $\ge$ into $\le$: $y \le -5$.

Answer. $y \le -5$ — solid dot at $-5$, shading left. The “or equal” bar survived the flip, so the dot stays solid. The habit of solving and graphing inequalities: re-read the sign after any negative divide — flip first, then draw.

Check. $y = -6$: $-4 \cdot (-6) = 24 \ge 20$, true. $y = -4$: $16 \ge 20$, false.

Skip the flip here and the arrow points the wrong way — the classic error in solving and graphing inequalities. Example 2 shows why solving and graphing inequalities rewards care over speed.

Problem. Solve the inequality and graph the solution.

Subtract 5 from both sides: $-2x \ge -4$. Divide by $-2$ and flip: $x \le 2$.

Answer. $x \le 2$ — solid dot at 2, shading left; solving and graphing inequalities marks it solid because $\le$ includes 2.

Check. $x = 1$ works: $5 - 2 = 3 \ge 1$, true. $x = 3$ fails: $5 - 6 = -1 \ge 1$, false.

Notice the flow of solving and graphing inequalities here: tidy, isolate, flip, check, then draw.

Two moves, one flip, one drawing — solving and graphing inequalities in miniature, and from here it is repetition with bigger numbers.

Tidy, isolate, flip if the divisor is negative, name the cutoff, draw the dot, pick the direction — five moves, one picture: that is solving and graphing inequalities. Solving and graphing inequalities never asks for more, and the three graphs above share this one rhythm.

1. Skipping the flip. $-4y \ge 20$ divided by $-4$ must end with $\le$; keep $\ge$ and the arrow points at the wrong half of the line.

2. Swapping solid and open. Mislabel the dot and the answer is off by one value.

3. Flipping when nothing demanded it. Only a true negative multiply or divide flips. Most lost marks in solving and graphing inequalities are flips that never needed to happen.

4. Dividing by a variable. From $bx < 3b$, dividing by $b$ is illegal until its sign is known: $b > 0$ gives $x < 3$, $b < 0$ gives $x > 3$.

Watch these four and solving and graphing inequalities stops costing points — every number line you draw says what you solved.