<< Chapter < Page Chapter >> Page >

Introduction

Now that you know how to solve quadratic equations, you are ready to learn how to solve quadratic inequalities.

Quadratic inequalities

A quadratic inequality is an inequality of the form

a x 2 + b x + c > 0 a x 2 + b x + c 0 a x 2 + b x + c < 0 a x 2 + b x + c 0

Solving a quadratic inequality corresponds to working out in what region the graph of a quadratic function lies above or below the x -axis.

Solve the inequality 4 x 2 - 4 x + 1 0 and interpret the solution graphically.

  1. Let f ( x ) = 4 x 2 - 4 x + 1 . Factorising this quadratic function gives f ( x ) = ( 2 x - 1 ) 2 .

  2. ( 2 x - 1 ) 2 0
  3. f ( x ) = 0 only when x = 1 2 .

  4. This means that the graph of f ( x ) = 4 x 2 - 4 x + 1 touches the x -axis at x = 1 2 , but there are no regions where the graph is below the x -axis.

Find all the solutions to the inequality x 2 - 5 x + 6 0 .

  1. The factors of x 2 - 5 x + 6 are ( x - 3 ) ( x - 2 ) .

  2. x 2 - 5 x + 6 0 ( x - 3 ) ( x - 2 ) 0
  3. We need to figure out which values of x satisfy the inequality. From the answers we have five regions to consider.

  4. Let f ( x ) = x 2 - 5 x + 6 . For each region, choose any point in the region and evaluate the function.

    f ( x ) sign of f ( x )
    Region A x < 2 f ( 1 ) = 2 +
    Region B x = 2 f ( 2 ) = 0 +
    Region C 2 < x < 3 f ( 2 , 5 ) = - 2 , 5 -
    Region D x = 3 f ( 3 ) = 0 +
    Region E x > 3 f ( 4 ) = 2 +

    We see that the function is positive for x 2 and x 3 .

  5. We see that x 2 - 5 x + 6 0 is true for x 2 and x 3 .

Solve the quadratic inequality - x 2 - 3 x + 5 > 0 .

  1. Let f ( x ) = - x 2 - 3 x + 5 . f ( x ) cannot be factorised so, use the quadratic formula to determine the roots of f ( x ) . The x -intercepts are solutions to the quadratic equation

    - x 2 - 3 x + 5 = 0 x 2 + 3 x - 5 = 0 x = - 3 ± ( 3 ) 2 - 4 ( 1 ) ( - 5 ) 2 ( 1 ) = - 3 ± 29 2 x 1 = - 3 - 29 2 x 2 = - 3 + 29 2
  2. We need to figure out which values of x satisfy the inequality. From the answers we have five regions to consider.

  3. We can use another method to determine the sign of the function over different regions, by drawing a rough sketch of the graph of the function. We know that the roots of the function correspond to the x -intercepts of the graph. Let g ( x ) = - x 2 - 3 x + 5 . We can see that this is a parabola with a maximum turning point that intersects the x -axis at x 1 and x 2 .

    It is clear that g ( x ) > 0 for x 1 < x < x 2

  4. - x 2 - 3 x + 5 > 0 for x 1 < x < x 2

When working with an inequality where the variable is in the denominator, a different approach is needed.

Solve 2 x + 3 1 x - 3

  1. 2 x + 3 - 1 x - 3 0
  2. 2 ( x - 3 ) - ( x + 3 ) ( x + 3 ) ( x - 3 ) 0 x - 9 ( x + 3 ) ( x - 3 ) 0
  3. We see that the expression is negative for x < - 3 or 3 < x 9 .

  4. x < - 3 o r 3 < x 9

Khan academy video on quadratic inequalities - 1

End of chapter exercises

Solve the following inequalities and show your answer on a number line.

  1. Solve: x 2 - x < 12 .
  2. Solve: 3 x 2 > - x + 4
  3. Solve: y 2 < - y - 2
  4. Solve: - t 2 + 2 t > - 3
  5. Solve: s 2 - 4 s > - 6
  6. Solve: 0 7 x 2 - x + 8
  7. Solve: 0 - 4 x 2 - x
  8. Solve: 0 6 x 2
  9. Solve: 2 x 2 + x + 6 0
  10. Solve for x if: x x - 3 < 2 and x 3 .
  11. Solve for x if: 4 x - 3 1 .
  12. Solve for x if: 4 ( x - 3 ) 2 < 1 .
  13. Solve for x : 2 x - 2 x - 3 > 3
  14. Solve for x : - 3 ( x - 3 ) ( x + 1 ) < 0
  15. Solve: ( 2 x - 3 ) 2 < 4
  16. Solve: 2 x 15 - x x
  17. Solve for x :     x 2 + 3 3 x - 2 0
  18. Solve: x - 2 3 x
  19. Solve for x : x 2 + 3 x - 4 5 + x 4 0
  20. Determine all real solutions: x - 2 3 - x 1

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 11 maths' conversation and receive update notifications?

Ask
Tess Armstrong
Start Quiz
JavaChamp Team
Start Quiz
Danielrosenberger
Start Quiz
Rachel Carlisle
Start Quiz
Yacoub Jayoghli
Start Quiz