Linear Inequations ICSE Class 8: Properties, Solution Sets and Number Lines

What Linear Inequations is about

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Linear Inequations sits in the Algebra theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Replacement set and solution set

• Replacement set: all the values the variable may take, given in the question.
• $\mathbb{N}=\{1,2,3,\dots \}$ (natural numbers)
• $\mathbb{W}=\{0,1,2,\dots \}$ (whole numbers)
• $\mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$ (integers)
• Solution set: those members of the replacement set that satisfy the inequation.

The same inequation can have different solution sets for different replacement sets. $x<4$ gives $\{1,2,3\}$ in $\mathbb{N}$ but $\{0,1,2,3\}$ in $\mathbb{W}$.

Worked example

Find the solution set of $x<4$ when $x\in \mathbb{N}$, and when $x\in \mathbb{W}$.

Try it first, then show the solutionSolution

1. Natural numbers less than 4: 1, 2, 3.
2. Whole numbers less than 4: 0, 1, 2, 3.

Answer: $\{1,2,3\}$; $\{0,1,2,3\}$

Properties of inequalities

For any numbers $a$, $b$, $c$ with $a<b$:
1. Adding or subtracting the same number keeps the sign: $a+c<b+c$ and $a-c<b-c$.
2. Multiplying or dividing by a positive number keeps the sign: if $c>0$, $ac<bc$.
3. Multiplying or dividing by a negative number reverses the sign: if $c<0$, $ac>bc$.

Why rule 3? $2<5$, but $-2>-5$ on the number line: multiplying by $-1$ flips the order.

Also, swapping the two sides reverses the sign: $a<b$ is the same as $b>a$.

Worked example

Solve $-3x>12$, $x\in \mathbb{Z}$.

Try it first, then show the solutionSolution

1. Divide both sides by $-3$ (negative), so reverse the sign.
2. $x<-4$.
3. Solution set $=\{\dots ,-7,-6,-5\}$.

Answer: $x<-4$: $\{\dots ,-7,-6,-5\}$

Solving a linear inequation

Steps (just like an equation):
1. Remove brackets and fractions (multiply by the LCM, a positive number).
2. Move the $x$ terms to one side and the numbers to the other, using rule 1.
3. Divide by the coefficient of $x$; flip the sign if it is negative.
4. List the members of the replacement set that fit.

You can avoid dividing by a negative by moving the $x$ terms to the side where the coefficient stays positive.

Worked example

Solve $3x-5\le 10$, $x\in \mathbb{W}$.

Try it first, then show the solutionSolution

1. Add 5: $3x\le 15$.
2. Divide by 3: $x\le 5$.
3. Whole numbers: $\{0,1,2,3,4,5\}$.

Answer: $\{0,1,2,3,4,5\}$

Showing the solution on a number line

When the replacement set is $\mathbb{N}$, $\mathbb{W}$ or $\mathbb{Z}$, the solutions are separate points.

Draw-and-label checklist:

• a straight line with equally spaced integers marked and labelled;
• a thick dot on each integer in the solution set;
• if the set goes on for ever, dots on the first few values and three dots (or an arrow) to show it continues;
• no dots on values that are excluded (for example the end of a strict $<$).

Worked example

Show $-2<x\le 3$, $x\in \mathbb{Z}$, on a number line.

Try it first, then show the solutionSolution

1. $-2$ is not included (strict $<$); 3 is included ($\le$).
2. Solution set $=\{-1,0,1,2,3\}$.
3. Mark integers from $-3$ to 4 and put thick dots on $-1,0,1,2,3$.

Answer: $\{-1,0,1,2,3\}$ shown as five dots

Inequations with fractions

Multiply every term on both sides by the LCM of the denominators. The LCM is positive, so the sign does not change.

Use brackets when multiplying a fraction with more than one term on top: $6\times \frac{x-2}{3}=2(x-2)$.

Worked example

Solve $\frac{x}{2}-1>\frac{x}{3}+1$, $x\in \mathbb{Z}$.

Try it first, then show the solutionSolution

1. Multiply by 6: $3x-6>2x+6$.
2. Subtract $2x$ and add 6: $x>12$.
3. Solution set $=\{13,14,15,\dots \}$.

Answer: $\{13,14,15,\dots \}$

Combined (double) inequations

An inequation like $-3\le 2x+1<7$ means both $-3\le 2x+1$ and $2x+1<7$.

You can work on all three parts at once, doing the same operation to each part. If you multiply or divide by a negative, both signs flip.

Alternatively, solve the two halves separately and take the values that satisfy both.

Worked example

Solve $-3\le 2x+1<7$, $x\in \mathbb{Z}$.

Try it first, then show the solutionSolution

1. Subtract 1 from all parts: $-4\le 2x<6$.
2. Divide all parts by 2: $-2\le x<3$.
3. Solution set $=\{-2,-1,0,1,2\}$.

Answer: $\{-2,-1,0,1,2\}$

Maths in real life: Linear Inequations

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Combined Inequations on the Number Line
• Solving Linear Inequations in One Variable

Linear Inequations formula sheet

Add or subtract

$a<b\Rightarrow a\pm c<b\pm c$

Sign unchanged.

Multiply by a positive

$a<b,c>0\Rightarrow ac<bc$

Same for division.

Multiply by a negative

$a<b,c<0\Rightarrow ac>bc$

Sign reverses; same for division.

Swap sides

$a<b\iff b>a$

Reading from the other side.

Number sets

$\mathbb{N}\subset \mathbb{W}\subset \mathbb{Z}$

N starts at 1, W at 0; Z includes negatives.

How the school exam asks Linear Inequations

The question patterns to expect, and the method that works for each.

Tricks to remember Linear Inequations

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Negative flips

  Multiply or divide by a negative, flip the sign. Add, subtract or use a positive, keep it.

  Use it when: Every inequation.

• The crocodile eats the bigger number

  The open mouth of < or > always faces the larger side.

  Use it when: Reading and writing the symbols.

• At most ≤, at least ≥

  'Most' is a ceiling (≤); 'least' is a floor (≥).

  Use it when: Word problems.

• N starts at 1

  Natural numbers start at 1, whole numbers at 0; integers go both ways.

  Use it when: Listing solution sets correctly.

Linear Inequations: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Linear Inequations

The slips examiners see most often, and what to write instead.

• ✗Mistake: Not reversing the sign

  Dividing by a negative number reverses the order.

  Wrong working

  $-3x>12\Rightarrow x>-4$

  Right working

  $-3x>12\Rightarrow x<-4$

  ✓Do this: Circle any negative divisor and flip straight away.

  Usually costs: The whole solution set.

• ✗Mistake: Including 0 for natural numbers

  $\mathbb{N}$ starts at 1.

  Wrong working

  $x<4$, $x\in \mathbb{N}$: $\{0,1,2,3\}$

  Right working

  $\{1,2,3\}$

  ✓Do this: Write the replacement set before listing.

  Usually costs: 1 mark.

• ✗Mistake: Including the end point of a strict inequality

  $x>4$ does not include 4.

  ✓Do this: Check the end value in the original inequation.

  Usually costs: 1 mark.

• ✗Mistake: Flipping only one sign in a combined inequation

  Dividing all three parts by a negative reverses both signs.

  Wrong working

  $-8\le -2x<2\Rightarrow 4\ge x<-1$

  Right working

  $4\ge x>-1$

  ✓Do this: Rewrite the whole line after flipping, smallest value on the left.

  Usually costs: 2 marks.

• ✗Mistake: Rounding the wrong way in word problems

  With 'at most', you round down; with 'at least', you round up.

  ✓Do this: Ask: does my rounded answer still satisfy the inequation?

  Usually costs: The final answer.

Practice questions on Linear Inequations

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Multiplying both sides of $3<5$ by $-2$ gives:
   (a) $-6<-10$
   (b) $-6>-10$
   (c) $6<10$
   (d) $-6=-10$

   Hint

   Negative multiplier.

   Answer

   (b)

2. Q2 · 2 marks · Short answerExam level

   Solve $x+4>9$, $x\in \{1,2,\dots ,10\}$.

   Hint

   Subtract 4.

   Answer

   $x>5$: $\{6,7,8,9,10\}$.

3. Q3 · 2 marks · Short answerExam level

   Solve $7-x\ge 3$, $x\in \mathbb{W}$.

   Hint

   $-x\ge -4$.

   Answer

   $x\le 4$: $\{0,1,2,3,4\}$.

4. Q4 · 3 marks · Short answerHarder exam level

   Solve $3(x+2)>5x-4$, $x\in \mathbb{N}$.

   Hint

   Expand, then collect.

   Answer

   $3x+6>5x-4\Rightarrow 10>2x\Rightarrow x<5$: $\{1,2,3,4\}$.

5. Q5 · 1 mark · Assertion-reasonHarder exam level

   Assertion (A): If $-2x<6$, then $x>-3$.
   Reason (R): Dividing both sides of an inequality by a negative number reverses the inequality sign.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   Divide by −2.

   Answer

   (c)

6. Q6 · 3 marks · Short answerHarder exam level

   Solve $\frac{x}{3}+2\ge \frac{x}{2}-1$, $x\in \mathbb{W}$.

   Hint

   Multiply by 6.

   Answer

   $2x+12\ge 3x-6\Rightarrow x\le 18$: $\{0,1,2,\dots ,18\}$.

7. Q7 · 3 marks · Short answerHarder exam level

   Solve $-5<3x-2\le 7$, $x\in \mathbb{Z}$, and show it on a number line.

   Hint

   Add 2 to all parts.

   Answer

   $-3<3x\le 9\Rightarrow -1<x\le 3$: $\{0,1,2,3\}$, shown as four dots.

8. Q8 · 2 marks · Short answerHarder exam level

   Find the greatest integer $x$ for which $4x+3<2x+14$.

   Hint

   Solve, then pick the largest integer.

   Answer

   $2x<11\Rightarrow x<5.5$; greatest integer 5.

9. Q9 · 3 marks · Short answerHarder exam level

   The sum of three consecutive natural numbers is less than 30. Find the greatest such three numbers.

   Hint

   Let them be $x$, $x+1$, $x+2$.

   Answer

   $3x+3<30\Rightarrow x<9$. Greatest $x=8$: the numbers are 8, 9, 10 (sum 27).

10. Q10 · 2 marks · Short answerHarder exam level

    Write the solution set of $x\ge -2$ and $x<3$, $x\in \mathbb{Z}$, as one combined inequation and as a list.

    Hint

    Both must hold.

    Answer

    $-2\le x<3$: $\{-2,-1,0,1,2\}$.

11. Q11 · 3 marks · Short answerHOTS / competency

    Solve $\frac{2x+1}{3}-\frac{x-1}{2}>1$, $x\in \{1,2,3,4,5,6\}$.

    Hint

    Multiply by 6, watch the minus before the bracket.

    Answer

    $2(2x+1)-3(x-1)>6\Rightarrow 4x+2-3x+3>6\Rightarrow x>1$: $\{2,3,4,5,6\}$.

12. Q12 · 4 marks · Long answerHOTS / competency

    A student scored 62, 75, 68 and 80 in four tests. What is the least score she needs in the fifth test so that her average is at least 72?

    Hint

    Average ≥ 72 means total ≥ 360.

    Answer

    $\frac{285+x}{5}\ge 72\Rightarrow 285+x\ge 360\Rightarrow x\ge 75$. Least score 75.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.What does ≥ mean?Show answer

   Greater than or equal to (at least).

2. 2.What happens to the sign when you multiply both sides by −1?Show answer

   It reverses.

3. 3.Write the whole numbers less than 3.Show answer

   {0, 1, 2}.

4. 4.Solve x + 5 > 8, x ∈ N, x ≤ 6.Show answer

   {4, 5, 6}.

5. 5.Is 4 a solution of 2x < 8?Show answer

   No (8 < 8 is false).

6. 6.Solve −x ≤ 2.Show answer

   x ≥ −2.

7. 7.What is the replacement set?Show answer

   The set of values the variable may take.

8. 8.Which set includes negative numbers: N, W or Z?Show answer

   Z.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Did you reverse the sign when dividing by a negative?
• ☐Did you check which end points are included?
• ☐Did you list only members of the replacement set?
• ☐Did you multiply every term by the LCM?
• ☐In a combined inequation, did you treat all three parts the same way?
• ☐In word problems, did you round in the right direction?

Linear Inequations: frequently asked questions