Inequalities

An equation says "exactly 49 yuan", but life talks more about "at least 1.2 m tall to ride" or "the bag won't hold more than 10 books". Sentences carrying "at least" and "no more than" are written in math as inequalities. Most of the solving moves are borrowed straight from Solving Linear Equations — with one place that needs special care, revealed below.

The four symbols

$a < b$ : a is less than b, so a sits to the left of b;
$a > b$ : a is greater than b, so a sits to the right of b;
$a \le b$ : a is at most b — smaller or exactly equal both pass;
$a \ge b$ : a is at least b — greater or exactly equal both count.

The pair with the equals sign ( $\le$ , $\ge$ ) is the easiest to fudge: "no more than 8 people" may be exactly 8; "more than 8" does not — you need 9 to qualify.

Drawing them on the number line

An inequality's answer is often not one number but a whole stretch of numbers, and the number line is its portrait:

$x > 3$ : every number to the right of 3. Draw an open circle at 3 (3 itself fails) and an arrow pointing right;
$x \le -1$ : −1 and everything to its left. Draw a closed dot at −1 (−1 qualifies) and an arrow pointing left.

Open or closed

With an equals sign ( $\le$ , $\ge$ ) the dot is closed — the endpoint is in; without one ( $<$ , $>$ ) the circle is open — the endpoint is out. One line to remember: equals means solid.

Solving like an equation, with one twist

Moving terms, adding or subtracting the same amount from both sides — all the old equation moves work:

$x + 2 > 5 \implies x > 3$

But the moment you multiply or divide both sides by a negative number, the sign must flip:

$-2x < 6 \implies x > -3$

Why? Negatives mirror the number line: originally $2 < 3$ ; after both change sign it becomes $-2 > -3$ . Flipping the world flips the less-than into a greater-than.

Check it to trust it

Solving $-2x < 6$ gave $x > -3$ . Test $x = 0$ (to the right of $-3$ ): the left side is $-2 \times 0 = 0$ , and indeed $0 < 6$ — it works. Then test $x = -4$ (to the left of $-3$ ): the left side is $-2 \times (-4) = 8$ , and $8 < 6$ fails — the rejected region deserved rejection.

Inequalities in everyday life

Roller coaster: height $h \ge 1.2$ m — exactly 1.2 m gets in;
Elevator: passenger count $n \le 10$ — the eleventh person waits for the next ride;
Phone battery: a warning fires when the charge $b \le 20\%$ .

Translating a sentence into an inequality

"Riders taller than 1.4 m buy an adult ticket": let height be $h$ metres. "Taller than" excludes 1.4 itself, so write $h > 1.4$ . If the rule were "1.4 m and above", it becomes $h \ge 1.4$ — one word apart, different symbols.

Check yourself

Quick quiz

1. How do you write “no more than 8 people” in symbols?
2. Solve -2x < 6. What do you get?
3. On a number line, how is the endpoint of x ≥ 1 drawn?