Lesson 10 of 10 · 14 min

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Inequalities and graphing regions, full chapter
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Must-know facts

16 facts

1Inequality signs: < less than, > greater than, ≤ less than or equal to, ≥ greater than or equal to.
2A solution makes the inequality true; the solution set collects all solutions.
3Adding or subtracting the same number on both sides keeps the sign.
4Multiplying or dividing by a positive number keeps the sign.
5Multiplying or dividing by a negative number reverses the sign.
6Clear fractions by multiplying by the (positive) LCM of the denominators.
7Open circle and round bracket: end excluded (< or >).
8Filled circle and square bracket: end included (≤ or ≥).
9∞ and −∞ always take round brackets.
10In a double inequality, do each step to all three parts.
11A system's solution is the intersection of the separate solution sets; it may be empty.
12The solution set depends on whether x is natural, integer or real.
13At most → ≤; at least → ≥.
147x − 4 < 4x + 8 gives x < 4; 2x + 5 > 5x − 10 gives x < 5.
15−3 ≤ 2x − 5 < 7 gives 1 ≤ x < 6.
1645x ≤ 500 gives at most 11 cartons.

Common traps

Where marks are lost

Dividing −3x > −15 by −3 and writing x > 5.

Dividing by a negative number reverses the sign: x < 5. Check with x = 4: −12 > −15 is true.

Drawing a filled circle for x < 4.

4 is not a solution of x < 4, so the circle is open; filled circles are only for ≤ and ≥.

Writing [2, 7] for 2 ≤ x < 7.

The right end is strict, so it takes a round bracket: [2, 7).

Reversing only one sign in −4 < −2x ≤ 2 when dividing by −2.

Both signs reverse: 2 > x ≥ −1, which is −1 ≤ x < 2.

Multiplying 1/x < 2 by x to get 1 < 2x.

The sign of x is unknown, so the direction is unknown; split into x > 0 and x < 0 or move everything to one side first.

Opening −2(x − 1) as −2x − 2.

−2 × (−1) = +2, so −2(x − 1) = −2x + 2.

Answering '11.1 cartons' for 45x ≤ 500.

Cartons are whole numbers; the largest allowed value is 11.

Taking the union instead of the intersection for a system.

A system needs every inequality at once, so keep only the common values.

Formulas

11 to know

Rule 1

a < b ⇒ a + c < b + c and a − c < b − c

True for every real c; the same for ≤, >, ≥.

Rule 2 (positive)

a < b and k > 0 ⇒ ka < kb and a/k < b/k

Sign kept.

Rule 2 (negative)

a < b and k < 0 ⇒ ka > kb and a/k > b/k

Sign reversed.

Solving ax + b < c

x < (c − b)/a if a > 0; x > (c − b)/a if a < 0

The same pattern for ≤, >, ≥.

Open intervals

(a, b) = {x : a < x < b}

Both ends excluded.

Closed interval

[a, b] = {x : a ≤ x ≤ b}

Both ends included.

Half-open intervals

[a, b) = {x : a ≤ x < b}; (a, b] = {x : a < x ≤ b}

Bracket type follows each end.

Rays

(−∞, a) = {x : x < a}; [a, ∞) = {x : x ≥ a}

∞ always takes a round bracket.

Double inequality

p < ax + b < q, a > 0 ⇒ (p − b)/a < x < (q − b)/a

If a < 0 both signs reverse.

Average condition

(m₁ + m₂ + x)/3 ≥ A ⇒ x ≥ 3A − m₁ − m₂

Minimum mark needed for an average of at least A.

Temperature scales

F = (9/5)C + 32; C = (5/9)(F − 32)

Both increase together, so ranges map without reversing.

Key terms

12 terms

Inequality: A statement comparing two numbers or expressions with <, >, ≤ or ≥.
Strict inequality: One using < or >, which excludes the boundary value.
Slack inequality: One using ≤ or ≥, which includes the boundary value.
Linear inequality: An inequality in which the variables appear only to the first power, such as ax + b < 0 with a ≠ 0.
Double inequality: A statement such as a < x < b that places a quantity between two others.
Solution: A value of the variable that makes the inequality true.
Solution set: The set of all solutions of an inequality or system.
Interval: A stretch of the real line, written with round or square brackets.
Open circle: The number-line mark for an excluded end point.
Filled circle: The number-line mark for an included end point.
System of inequalities: Two or more inequalities to be satisfied at the same time.
Empty set: The solution set of a system with no common value, written ∅.