Mathematics notes, chapter 7

Linear inequalities

Textbook Mathematics Form 1 (DLP), pages 148–166

What this chapter covers

Compare numbers with the symbols > and <, describe an inequality, show it on a number line and form an algebraic inequality (7.1)
Use the converse and transitive properties, the additive and multiplicative inverses, and know when +, −, × and ÷ keep or reverse the inequality symbol (7.1)
Form linear inequalities from daily life situations, and write a situation for a given inequality (7.2)
Solve linear inequalities in one variable and use them to solve problems (7.2)
Solve simultaneous linear inequalities in one variable using a number line (7.2)

7.1Inequalities

An inequality is the relationship between two quantities that do not have the same value. We compare them with the symbols <, >, ≤ and ≥.

In daily life we compare prices, masses, temperatures and heights. A phone that costs RM1 400 is cheaper than one that costs RM1 600, so 1 400 < 1 600.

Symbol	Meaning	Example
<	is less than	−2 < 3
>	is greater than	−2 > −7
≤	is less than or equal to	v ≤ 110
≥	is greater than or equal to	t ≥ 9

The symbol always opens towards the bigger number, like a hungry mouth: 3 > −2 and −2 < 3 say the same thing.

Comparing numbers on a number line

Two number lines: −2 and 3 marked on a line from −3 to 4; −7 and −2 marked on a line from −8 to −1 — The number further to the **right** is the greater number (textbook p. 150).

Self Practice 7.1a (textbook p. 151): fill in < or >

(a) −6 ☐ 0 (b) 10 cm ☐ 50 mm (c) 1 200 g ☐ 1.6 kg
(a) −6 is to the left of 0, so −6 < 0. "−6 is less than 0."
(b) Change to the same unit first: 10 cm = 100 mm. 100 mm > 50 mm, so 10 cm > 50 mm.
(c) 1.6 kg = 1 600 g. 1 200 g < 1 600 g, so 1 200 g < 1.6 kg.

Comparing 10 cm with 50 mm by looking only at the numbers (10 < 50). Always change both quantities to the same unit first.

Describing an inequality and forming an algebraic inequality

A number line with 4 marked and x marked to the right of 4 — x lies to the right of 4: "x is greater than 4", written x > 4.

The road sign on the highway says the speed limit is 110 km/h. If v is the speed, then v = 110 or v < 110 is allowed. We join them with one symbol: v ≤ 110. The cars that break the limit have v > 110.

Textbook pp. 152–153. v can be any number, including decimals and fractions.

Four number lines from −1 to 5 showing x > 2, x ≥ 2, x < 2 and x ≤ 2 — Open circle: the number itself is **not** included (<, >). Solid circle: the number is included (≤, ≥).

Example 3 (textbook p. 153): overtime pay

Fatimah earns overtime pay when she works at least 9 hours a day. Let t be her working hours per day.
(a) t is greater than or equal to 9.
(b) Draw a solid circle at 9 (9 hours is included) and an arrow to the right.

The algebraic inequality is t ≥ 9.

Using an open circle for ≤ or ≥. "At least 9" includes 9, so the circle must be solid (shaded).

(a) A sign shows that a lorry's load, m tonnes, must be at most 8 tonnes. Write the inequality.
(b) Fatimah did not earn any overtime pay. Write an inequality for her working hours, t.

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(a) m ≤ 8 (solid circle at 8, arrow to the left)
(b) t < 9 (open circle at 9, arrow to the left)

Properties of inequalities

Property	Rule	Example (textbook p. 154)
Converse	If a < b, then b > a.	10 < 24, so 24 > 10 −4 > −13, so −13 < −4
Transitive	If a < b < c, then a < c.	−5 < −1 < 9, so −5 < 9

The converse just reads the same inequality from the other side. Swap the two numbers and turn the symbol round.

Inequalities and the four operations

Start with 2 < 6 and do the same thing to both sides (textbook p. 156).

Do this to both sides	Result	Symbol
Add 3	5 < 9	stays the same
Subtract 3	−1 < 3
Add (−3)	−1 < 3
Subtract (−3)	5 < 9
Multiply by 3	6 < 18
Divide by 2	1 < 3
Multiply by (−3)	−6 > −18	reversed
Divide by (−2)	−1 > −3
Multiply by (−1) (additive inverse)	−2 > −6
Take the reciprocal (multiplicative inverse)	12 > 16

Number line from −8 to 8. Dashed arcs join 2 to −2 and 6 to −6. 2 < 6 but −2 > −6 — Multiplying by a negative number flips the numbers to the other side of 0, so their order is reversed.

Remember: adding or subtracting any number keeps the symbol. Multiplying or dividing by a positive number keeps the symbol. Multiplying or dividing by a negative number reverses the symbol.

Thinking that subtracting a negative number reverses the symbol. Only multiplying or dividing by a negative number does. 2 − (−3) < 6 − (−3) is 5 < 9.

Given 6 < 12, fill in < or >.
(a) 6 × (−3) ☐ 12 × (−3) (b) 16 ☐ 112 (c) −6 ☐ −12 (d) Write the converse of 8 > −8 (e) Write the transitive property of −15 < −8 < 0

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(a) −18 > −36, so >
(b) > (reciprocal reverses)
(c) > (× (−1) reverses)
(d) −8 < 8
(e) −15 < 0

7.2Linear inequalities in one variable

A linear inequality in one variable is an inequality with one variable whose power is 1, such as 3x < 7 or y − 4 > 5 + 2y.

Forming linear inequalities from situations

Words in the question	Symbol
more than, exceeds, above	>
less than, below, fewer than	<
at least, not less than, minimum, … and above	≥
at most, not more than, maximum, cannot exceed	≤

Example 5 (textbook p. 158)

(a) Pak Samad takes less than 42 days to make a gasing uri: t < 42
(b) A fish must be at least 32 cm long to win a prize: l ≥ 32
(c) A cake of not more than 2 kg is cut into 10 equal slices of x kg: 10x ≤ 2
(d) Mr Mohan donates 3% of his profit p and wants the donation to exceed RM240: 3100 p > 240

Example 6 (textbook p. 159): from inequality to situation

(a) h ≥ 110, h = height in cm to ride a roller coaster: a rider must be at least 110 cm tall.
(b) T < −5, T = freezer temperature in °C: the freezer is colder than −5°C.
(c) m > 4 600, m = Mr Siva's monthly salary in RM: he earns more than RM4 600.

Writing "at least 32" as l > 32. "At least" means 32 is allowed, so use ≥.

Write a linear inequality.
(a) A terrace house costs RMx, which is RM450 000 and above.
(b) The passing mark is 50. Hajar got y marks and failed.
(c) k people in 5 debating teams; each team cannot have more than 6 people.

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(a) x ≥ 450 000
(b) y < 50
(c) k ≤ 5 × 6, so k ≤ 30

Solving linear inequalities

Solve an inequality the same way as a linear equation: get the variable alone. The only new rule is: when you multiply or divide by a negative number, reverse the symbol.

Example 7 (textbook p. 160)

(a) x − 2 ≤ 6
    x − 2 + 2 ≤ 6 + 2 (add 2)
    x ≤ 8
(b) 7x ≥ 28
    7x7 ≥ 287 (divide by 7, a positive number)
    x ≥ 4
(c) −x3 < 9
    −x3 × (−3) > 9 × (−3) (multiply by −3 and reverse)
    x > −27
(d) 7 − 4x > 15
    7 − 4x − 7 > 15 − 7 (subtract 7)
    −4x > 8
    −4x−4 < 8−4 (divide by −4 and reverse)
    x < −2

An inequality has many solutions. Check one value: in (d) try x = −3: 7 − 4(−3) = 19, and 19 > 15. ✓

Dividing −4x > 8 by −4 and writing x > −2. You must reverse the symbol: x < −2.

Solve. (a) x + 3 ≥ 10 (b) −2x < 18 (c) −7 > x3 (d) 16 − 5x ≤ −4

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(a) x ≥ 10 − 3, x ≥ 7
(b) x > 18 ÷ (−2), x > −9
(c) −21 > x, so x < −21
(d) −5x ≤ −20, x ≥ 4

Solving problems

Example (textbook p. 161): books at RM12.50 each

Ghani spends not more than RM80 on books that cost RM12.50 each. Find the maximum number of books he can buy.
Let n = number of books.
12.5n ≤ 80
n ≤ 8012.5
n ≤ 6.4

n is a whole number, so the maximum number of books is 6.
Check: 6 × RM12.50 = RM75 (≤ RM80) ✓; 7 × RM12.50 = RM87.50 (> RM80) ✗

When the answer must be a whole number (books, people, months): for a maximum, round down; for a minimum, round up to the next whole number that works.

Fatimah is paid 10 sen for each canned drink she sells. She wants to earn at least RM20 an hour. How many cans must she sell in an hour?

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0.10n ≥ 20
n ≥ 200
At least 200 cans an hour.

Simultaneous linear inequalities

The solution of simultaneous linear inequalities is the set of values that satisfy both inequalities: the common region on a number line.

Daily sugar intake should be between 25 g and 37.5 g: m > 25 and m < 37.5, or 25 < m < 37.5 (textbook p. 162).

Example 8(a) (textbook p. 162)

Solve 2x + 5 < 11 and 3x − 10 < 5.
2x + 5 < 11 → 2x < 6 → x < 3
3x − 10 < 5 → 3x < 15 → x < 5

Both arrows cover the region to the left of 3. The solution is x < 3.

Example 8(b) (textbook p. 163)

Solve 8x + 5 ≥ 5x − 13 and 3x − 4 > 9x + 20.
8x − 5x ≥ −13 − 5 → 3x ≥ −18 → x ≥ −6
3x − 9x > 20 + 4 → −6x > 24 → x < −4 (divide by −6, reverse)

The solution is −6 ≤ x < −4.

Example: listing the integer solutions

Solve x + 4 > 6 and 3x ≤ 18, then list the integers that satisfy both.
x > 2 and x ≤ 6, so 2 < x ≤ 6.

Integers: 3, 4, 5, 6 (2 is not included; 6 is included).

If the two arrows point away from each other and do not overlap, there is no solution. For example, x < 2 and x > 5 have no common values.

No common region, so no solution.

Writing the answer for x > 2 and x > 5 as 2 < x < 5. Both arrows point right, so the common region is x > 5 only. Always draw the number line.

Solve. (a) x + 2 > 4 and 4x > 20 (b) x − 3 ≤ −6 and 3x − 4 ≤ 5 (c) 5x − 3 < 2 and 4x + 6 ≥ x + 3

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(a) x > 2 and x > 5, so x > 5
(b) x ≤ −3 and x ≤ 3, so x ≤ −3
(c) x < 1 and 3x ≥ −3, x ≥ −1, so −1 ≤ x < 1

Summary

Remember	Key fact
Symbols	< less than, > greater than, ≤ less than or equal to, ≥ greater than or equal to
Number line	Open circle for < and >; solid circle for ≤ and ≥
Converse	If a < b, then b > a
Transitive	If a < b < c, then a < c
+ and −	If a < b, then a + c < b + c and a − c < b − c (symbol stays)
× and ÷ by positive c	a × c < b × c and ac < bc (symbol stays)
× and ÷ by negative (−c)	a × (−c) > b × (−c) and a−c > b−c (symbol reverses)
Inverses	If a < b, then −a > −b; for positive a and b, 1a > 1b
Word clues	at least, minimum → ≥; at most, maximum, not more than → ≤
Simultaneous	Solve each one, draw both on one number line, take the common region