CMV6120Foundation Mathematics

Unit 5 : Linear and quadratic inequalities in one variable

Learning Objectives

Students should be able to :

State the basic properties of inequalities.

Solve a linear inequality in one variable.

Represent the solution graphically

Solve compound inequalities involving ‘and’ , ‘or’.

Solve a quadratic inequality in one variable.

Solve simple applied problems involving inequalities.

Activities

Teacher demonstration and students hand-on exercise.

Reference

Canotta 5A

Chapter 3

Linear and quadratic inequalities in one variable

1Basic properties of inequalities

For any three real numbers a, b and c:

Property / Example
1. If ab and bc, then ac. / 9>5 and 5>2, then 9>2
2. If ab, then a+cb+c. / 3>1, then 3+5>1+5
3. If ab, then
acbc when c>0;
acbc when c<0. / 8>6, then
8(2)>6(2) for 2>0;
8(-2)<6(-2) for –2<0.
4. If ab (where ab>0), then
/ 4>2, then

5. If, then . / , then .

Class practice:

If a < 0, then a2 ___ 0 / If a > 3, then a +4 > ___
If a < 3, then a -1 ___ 2 / If 2a >3, then a > ___
If -3a < -6, then a ___ 2 / If a < -6, then a ___ 12
If a > 3, then ___ / If a < -4, then ___

2How to solve Linear inequality in one variable

  1. Express the inequality as ax (<, ,>, ) b. (Use the technique of unit 1)
  2. Use property 3 to solve the inequality

E.g. 1 / Solve / Explanation
Solution / Property 3
E.g. 2 / Solve / Explanation
Solution / Property 3

3Graphical Representations

Solution / Graph / Remark
/ Put variable x on LHS.
Put number on RHS.
Arrow is in the same direction as the
Inequality sign.
Means
-25 is not included in the solution.
Means
1 is included in the solution.

Class practice:

Inequality / Graphical Representation
or

4Compound inequalities involving ‘and’ , ‘or’

Steps:

Solve each inequality separately.

Represent each solution graphically.

Shape the required region.:
AND : The common region will give the solution of x.
OR : Any region marked in step (2) will make up the solutions of x.

State the final answer.

* usually, the answer can be simplified so that it does not contain “AND” or “OR”.

E.g. 3 / Solve and / Explanation
Solution / Solve each inequality separately
Represent the solution graphically and shading.
State the final answer.
E.g. 4 / Solve and / Explanation
Solution / Solve each inequality separately
represent the solution graphically and shading.
State the final answer.
E.g. 5 / Solve / Explanation
Solution / The question is another style of writing an “AND” inequality.
There is no common region.
State the final answer.
E.g. 6 / Solve or / Explanation
Solution
State the final answer.
E.g. 7 / Solve or / Explanation
Solution
E.g. 8 / Solve or / Explanation
Solution
Further simplification not possible.

5Quadratic inequality in one variable.

Steps:

(1)Express the inequality as “(<, ,>, ) 0” with a > 0.

(2)Find the roots of and sketch the graph of .

No real roots
/ Double root / Two distinct roots

(3)Read the answer from the graph.

Class practice: use step 3 to find the value of x of the following.

x2 – 3x >0


Ans: x < 0 or x > 3 / x2 – 6x +5 < 0



Ans: 1 < x < 5 / -x2 + 6x + 7 > 0



Ans: -1 < x < 7
E.g. 9 / Solve . / Explanation
Solution
E.g. 10 / Solve . / Explanation
Solution / Note that there is no real root as the discriminant is less than 0.
None of the curve is below 0.

Unit 5: Linear and quadratic Inequalities in one variablePage 1 of 7