Grade 10 / inequalities 104 exercises (100% corrected)

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E11−1 E21−1 E3−11 E4−11 -8-7-6-5-4-3-2-1012345678 40 −10 −20 −10 ChingQuizz : 2 exercises available for Quizz assessment : 1. Introduction to inequalities E.844 Say whether the number 2 is a solution of the following inequalities: a 3 x > 5 b − 2 x > − 3 b 3 x + 7 5 x + 1 c 2( x + 1) > 7 E.9867 Say whether the number 2 is a solu-tion of the following inequations : a 3 x + 1 > 5 b − 2 x + 6 2 c 8(1 − x ) < − 5( x + 1) d (3 x − 8) 2 > − 3 E.2092 Which of the following inequalities hold true for x =2 : a 3 x + 2 > 5 b 5 − 2 x < − 1 c 2 x + 1 > x + 5 d ( x + 1)( x − 3) < 0 E.9868 Which of the following inequalities hold for x =2 : a x 2 − 3 x + 4 < 0 b x + 1 x − 5 > 0 E.2089 The following comparison is consid-ered : 3 x − 2 4 2 − x 2 Check whether the following numbers verify this inequality: a x = − 2 b x =1 c x =3 2. Representation of an inequality E.4124 Consider the six inequalities below : a x > − 1 b x < 1 c x > 1 d − x > 1 e − x < − 1 f − x > − 1 Associate each of these inequations, associate the set of its solutions has one of the representations below : E.6433 1 Hatch on the graduated line, the set of numbers x veri-fying the inequality x +3 > 2 : 2 Hash on the graduated line, the part of the numbers x verifying the comparison x − 1 < 4 : 3 Hash on the graduated line, the part of the numbers x verifying the comparison 2 × x> 4 : 4 Hash on the graduated line, the part of the numbers x verifying the comparison − x> 1 : E.5553 Justify that each of the statements below is false : a The inequation 3 x< 10 admits for solution the set of so-lutions represented on the scaled line below : b The inequation − 2 x> 5 admits for solution the set of so-lutions represented on the graduated line below : c The inequation x +3 < 4 admits for solution the set of so-lutions represented on the graduated line below : d The inequation − x +5 < 8 admits for solution the set of solutions represented on the graduated line below : E.4123 Consider the eight inequalities be-low : a x +2 > 4 b x +6 < 4 c 2 x> − 4 d 3 x< − 6 e 2 x +1 > 5 f − x< 2 g − 2 x< − 4 h − 0.5 x +1 > 0 Each of these inequations has one and only one of the follow-ing sets of numbers as its solution set : E 1 : ˇ all numbers strictly greater than 2 ı. E 2 : ˇ all numbers strictly less than 2 ı. E 3 : ˇ all numbers strictly greater than − 2 ı. E 4 : ˇ all numbers strictly less than − 2 ı. https://chingmath.fr chapExoCorrec/844 sacados/844 chapExoCorrec/9867 sacados/9867 chapExoCorrec/2092 sacados/2092 chapExoCorrec/9868 sacados/9868 chapExoCorrec/2089 sacados/2089 chapExoCorrec/4124 sacados/4124 E11−1 E21−1 E3−11 E4−11 chapExoCorrec/6433 sacados/6433 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 chapExoCorrec/5553 sacados/5553 40 −10 −20 −10 chapExoCorrec/4123 sacados/4123
-8-7-6-5-4-3-2-1012345678AB -8-7-6-5-4-3-2-1012345678CD -9-8-7-6-5-4-3-2-10123456789BA -9-8-7-6-5-4-3-2-10123456789CD -9-8-7-6-5-4-3-2-10123456789BA -9-8-7-6-5-4-3-2-10123456789CD 3. Inequality and operations E.11005 1 On the graduated right side, we have A − 2 and B 5 : a Compare the x-coordinates of these points : x A : : : x B b Place the two points A and B defined by: A x A + 2 ; B x B + 2 Compare the x-coordinates of these two points. 2 On the graduated line, we have C − 2 and D 5 : a Compare the x-coordinates of these points : x C : : : x D b Place the two points C and D defined by: C x C − 3 ; D x D − 3 Compare the x-coordinates of these two points. E.11014 1 On the graduated right side, we have A 1 and B 3 : a Compare the x-coordinates of these points : x A : : : x B b Place the two points A and B defined by: A 2 x A ; B 2 x B Compare the x-coordinates of these two points. 2 On the graduated line, we have C − 8 and D 4 : a Compare the x-coordinates of these points : x C : : : x D b Place the two points C and D defined by: C x C 4 ; D x D 4 Compare the x-coordinates of these two points. E.11348 1 On the graduated right side, we have A − 7 and B 3 : a Compare the x-coordinates of these points : x A : : : x B b Place the two points A and B defined by: A − x A ; B − x B Compare the x-coordinates of these two points. 2 On the graduated line, we have C 4 and D 6 : a Compare the x-coordinates of these points : x C : : : x D b Place the two points C and D defined by: C x C − 2 ; D x D − 2 Compare the x-coordinates of these two points. E.5665 Consider a number x indeterminate : a Si 2 x> 4 then x : : : b Si − x> 4 then x : : : c Si 5 x> 5 then x : : : d Si − 2 x> 6 then x : : : 4. First approaches E.8063 Solve the following inequalities and represent their solution set on a number line: a x + 1 > 0 b 2 x + 4 3 c − x + 3 5 Hint: Represent the set of solutions on a number line and in interval form. E.431 Solve the following inequalities and express their solution sets as intervals : a 3 x +1 >x +2 b 2 x +4 4 x − 1 c 5 x +3 4 x Hint: The solution sets will be represented on a number line and as intervals. E.9219 Solve the following inequalities: a 2 x − 1 5 x + 4 b 3 x + 2 <x + 2 Hint: The set of solutions will be represented on a gradu-ated line and in interval form. E.2455 Solve the following inequalities and represent the set of solu-tions on a number line in each case : a 6 x − 3 > 2 x − 9 b x + 2 2 x + 1 Hint: Represent the set of solutions on a number line and in interval form. E.9869 Solve the following inequalities and represent the set of solutions on a number line in each case : a 7 x + 3 > 4 x + 1 b 2 x + 1 8 x + 5 Hint: Represent the set of solutions on a number line and in interval form. E.11349 Solve the following inequalities and represent the set of solutions on a number line in each case : a 5 x + 1 < 3 x + 9 b 3 x + 2 < 5 x + 5 Hint: Represent the set of solutions on a number line and in interval form. https://chingmath.fr chapExoCorrec/11005 sacados/11005 -8-7-6-5-4-3-2-1012345678AB -8-7-6-5-4-3-2-1012345678CD chapExoCorrec/11014 sacados/11014 -9-8-7-6-5-4-3-2-10123456789BA -9-8-7-6-5-4-3-2-10123456789CD chapExoCorrec/11348 sacados/11348 -9-8-7-6-5-4-3-2-10123456789BA -9-8-7-6-5-4-3-2-10123456789CD chapExoCorrec/5665 sacados/5665 chapExoCorrec/8063 sacados/8063 chapExoCorrec/431 sacados/431 chapExoCorrec/9219 sacados/9219 chapExoCorrec/2455 sacados/2455 chapExoCorrec/9869 sacados/9869 chapExoCorrec/11349 sacados/11349
-8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 E.11350 Solve the following inequalities: a 2 x + 4 < 5 x − 7 b 3 x + 1 <x − 5 Hint: The set of solutions will be represented on a gradu-ated line and in interval form. E.11410 Solve the following inequalities: a 3 x + 3 < 6 x + 8 b − 3 x + 2 − 8 x − 7 Hint: The set of solutions will be represented on a gradu-ated line and in interval form. E.11467 Solve the following inequalities: a 4 x + 1 < 11 x + 10 b − 4 x + 3 − 12 x − 6 Hint: Give the set of solutions in the form of an interval. E.8032 Solve the inequalities below and give the set of solutions as an interval: a x + 1 > 0 b 2 x 4 c x + 2 5 d 3 x + 2 < − 1 5. Parts of R and literal expressions E.6432 1 Shade the part of the numbers on the right-hand scale that satisfy the comparison x> 2 : Give the notation for this set in the form of an interval. 2 Shade the part of the numbers on the number line that satisfy the comparison x< 4 : Give the notation for this set in the form of an interval. E.5666 Consider a number x undetermined but known to be strictly greater than 2 : x> 2 1 Hash the part of the graduated line below o where the number x is located : 2 Hash the part of the graded line below o where the num-ber 2 x is located : 3 Hash the part of the graded line below o where the num-ber x − 3 is located : 4 Hash the part of the graded line below où the number − 1 2 x : E.9226 Consider a number x that is undeter-mined but is known to verify the following frame : 2 <x< 4 1 Hash the part of the graduated line below o where the number x is located : 2 Hash the part of the graded line below o where the num-ber 2 x is located : 3 Hash the part of the scaled line below o where the number x − 3 is located : 4 Hash the part of the graded line below où the number − 1 2 x : 6. Simple equations E.845 Represent on a graduated line the solutions of the following equations : a x > 5 b x − 1 c 2 x > 2 d 3 x + 1 < − 2 e − x > 5 f − x + 1 > 3 E.847 french Polynesia - September 2005 Consider the inequation : 2 x − 5 3 − 11 x . 1 a Is the number 0 a solution to this inequation? Jus-tify the answer. b Is the number 1 a solution to this inequation? Justify the answer. 2 a Solve the inequation : 2 x − 5 3 − 11 x b Represent the solutions on a graduated line. https://chingmath.fr chapExoCorrec/11350 sacados/11350 chapExoCorrec/11410 sacados/11410 chapExoCorrec/11467 sacados/11467 chapExoCorrec/8032 sacados/8032 chapExoCorrec/6432 sacados/6432 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 chapExoCorrec/5666 sacados/5666 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 chapExoCorrec/9226 sacados/9226 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 -8-7-6-5-4-3-2-1012345678 chapExoCorrec/845 sacados/845 chapExoCorrec/847 sacados/847
E.344 Which of the following inequations accept the number 9 as a solution : a − 3 x + 2 0 b 5( x + 9) > 0 c x + 1 4 − 3 × x − 2 3 d 2 > x E.478 Solve the following inequalities and give their solution sets in interval form : a 2 x + 1 3 x − 1 b − x − 1 2 x + 2 E.9792 Solve the following inequalities and give the set of solutions in the form of intervals : a − 3 x + 7 x + 2 b − 6 x + 1 > 0 c − x 4 < 5 d − 3( x + 5) <x + 5 e − 3 x + 7 9 − x 7. Inequations and algebraic operations E.9224 Solve the following inequalities and plot the solutions on a graduated line: a 2 × (5 x − 1) + 2 3 x + 2 b 3(2 x + 7) x + 1 E.848 Solve the following inequalities and plot the results on a graduated line: a 3 x − (2 x + 4) 3 + 2( x + 1) b 3( − x + 1) − 4(2 x − 4) 5 E.853 Solve the following inequalities and plot the solutions on a graduated line: a 3(2 x + 1) > 3 − 3 × ( x − 5) E.857 Solve the following inequalities and graph the solutions : a 2( x + 3) − 3( x + 1) < 2(3 x + 1) b 2 × ( x + 8) 3 − 3 × (8 − 2 x ) E.850 Solve the following inequalities: a 3 x + 2(5 − x ) − 2 x + 1 b 214(3 x − 5) > 214(2 x + 1) E.11411 Solve the inequality: 3 2 x + 1 − 5 4 x − 2) 2 x + 1 Hint: The set of solutions will be represented on a number line and in interval form. E.11468 Solve the inequality: 4 5 x + 2 − 2 5 − x > 4 x + 5 Hint: Give the set of solutions in the form of an interval. E.9223 Solve the inequality below and plot the solutions on a number line: ( x − 2)(2 x − 4) 2 x 2 − 8 x + 1 E.846 Solve the following inequalities and represent their solution sets using a graduated line. a ( x + 1) 2 + 4 x 2 + 1 b (3 x − 2) 2 > 9 x 2 + 5 E.2512 Solve the following inequalities and plot the solutions on a graduated line: a (2 x − 3) 2 > (4 x + 1)( x + 5) 8. Inequalities and rational numbers E.856 Consider the expression : D = 4 x +2 5 . 1 Calculate D for x = 3 4 . Is the number 3 4 a solution of the inequation 4 x +2 5 < 3 ? 2 Solve the inequation 4 x +2 5 < 3 and plot the solutions on a graduated line? E.852 We wish to solve the inequation : 2 x +1 4 +1 > 2 x + x 2 1 Simplify the two expressions : 4 × 2 x + 1 4 + 1 ; 4 × 2 x + x 2 2 To solve this inequation, multiply each member of the inequation by 4 . E.9225 Solve the inequation : 3 x + 1 6 > 5 x − 3 8 E.9220 Solve the following inequalities and plot the results on a graduated line: a 2 x + 1 4 + 1 > 2 x + x 2 b x + 1 4 + 1 3 x 6 https://chingmath.fr chapExoCorrec/344 sacados/344 chapExoCorrec/478 sacados/478 chapExoCorrec/9792 sacados/9792 chapExoCorrec/9224 sacados/9224 chapExoCorrec/848 sacados/848 chapExoCorrec/853 sacados/853 chapExoCorrec/857 sacados/857 chapExoCorrec/850 sacados/850 chapExoCorrec/11411 sacados/11411 chapExoCorrec/11468 sacados/11468 chapExoCorrec/9223 sacados/9223 chapExoCorrec/846 sacados/846 chapExoCorrec/2512 sacados/2512 chapExoCorrec/856 sacados/856 chapExoCorrec/852 sacados/852 chapExoCorrec/9225 sacados/9225 chapExoCorrec/9220 sacados/9220
x1;5cm2;5cm5cmABCDPMNRR1R2 E.9221 Solve the following inequations and plot the results on a graduated line: a 2 x + 1 4 + 1 2 x + x 2 b x + 1 3 + 2 − x 15 > 2 x + 7 5 E.9222 Solve the following inequalities and plot the solutions on a graduated line: a 3 x + 2 4 + 2 − x 3 < x + 1 12 b 5 x + 1 2 + 5 9 7 x 6 − 1 18 E.9816 Solve the following inequalities and give the set of solutions in the form of intervals : a 3 x + 3 1 b 3 x − 1 4 − 1 E.9793 Solve the following inequalities and give the set of solutions in the form of intervals : a x + 1 2 + x < 0 b x − 2 − 4 <x + 1 E.2859 Solve the inequation : 3 x − 2 6 + 1 3 1 4 + 5 2 x E.9794 Solve the following inequalities and give the solutions in the form of intervals : a x − 1 6 + x + 1 3 < 2 b x + x 2 − x 6 x + 1 3 + 2 x − 3 6 E.9815 Solve the following inequalities and give the set of solutions in interval form : a 1 2 − 1 2 · x 1 3 · x + 1 6 b 1 5 · x + 1 2 > x − 1 3 9. Problems E.851 1 a 60 is solution of inequation : 2.5 x − 75 > 76 . b Solve the inequation and plot the solutions on an axis. Hatch the part of the axis that does not correspond to the solutions. 2 During the summer months, an ice cream vendor noticed that he was spending 75 euros a week to make, on aver-age, 150 ice creams. Knowing that an ice cream is sold 2.50 euros, how many ice creams must he sell, at least, in the week to have a profit higher than 76 euros? We’ll explain the process. E.855 1 Solve the inequation : x +15 2 3 ( x +27) 2 A research office employs 27 computer scientists and 15 mathematicians. It is planned to hire the same number x of computer scientists and mathematicians. How many specialists of each kind must be hired so that the num- ber of mathematicians is at least equal to two-thirds the number of computer scientists? E.860 The company Alo proposes a telephone subscription of 98 F per month and 1.30 F per minute of communication. Lao Company offers a phone subscription of 95 F per month and 1.45 F per minute of talk time. We denote x the number of call minutes per month. 1 Express as a function of x the amount of an Alo bill, then the amount of a Lao bill. 2 For which monthly call durations is it worth choosing Alo? E.849 In 2005, a Paris metro ticket cost 1.40 e . While the monthly subscription ˇCarte Orangeı cost 50.40 e to circulate freely within downtown Paris. After how many journeys does the Orange card become worth-while? Justify your answer. 10. Problems and geometry E.2506 ABCD is a rectangle: DC = 5 cm ; BC =2.5 cm N is the point on segment [ AD ] such that : AN =1.5 cm . M is a point on segment [ AB ] . Note x the length of segment [ AM ] expressed in centimeters ( x is between 0 and 5) . AMPN and MBCR are rectangles noted R 1 and R 2 respec-tively. 1 a Express, as a function of x , the perimeter of R 1 . b Express, as a function of x , the perimeter of R 2 . 2 What are the values of AM for which the perimeter of R 2 is greater than or equal to the perimeter of R 1 ? https://chingmath.fr chapExoCorrec/9221 sacados/9221 chapExoCorrec/9222 sacados/9222 chapExoCorrec/9816 sacados/9816 chapExoCorrec/9793 sacados/9793 chapExoCorrec/2859 sacados/2859 chapExoCorrec/9794 sacados/9794 chapExoCorrec/9815 sacados/9815 chapExoCorrec/851 sacados/851 chapExoCorrec/855 sacados/855 chapExoCorrec/860 sacados/860 Clermont-Ferrand - 2000 chapExoCorrec/849 sacados/849 chapExoCorrec/2506 sacados/2506 x1;5cm2;5cm5cmABCDPMNRR1R2
11. Inequations and factoring E.473 Solve the following inequalities: a ( x + 1)(1 − x ) > (2 x − 1)( x + 1) b x 3 − x 0 c ( x + 1) 2 − ( x + 1)(2 − x ) 0 E.6688 Solve the inequation : (2 x − 1)(3 − x ) (2 x − 1)(5 x + 1) E.9791 Consider the inequation : (3 x − 1)(4 x +5) > 3(3 − 2 x )(2 − 6 x ) 1 Factor: (3 x − 1)(4 x +5) − 3(3 − 2 x )(2 − 6 x ) . 2 Solve the inequation ( E ) . E.6686 Solve the following inequalities: a (2 x − 1)(3 x + 1) (4 − x )(3 x + 1) b (3 x + 2)(2 − 3 x ) (3 x + 2)(5 x − 2) E.9807 Solve the inequation : (3 − x )(2 x +3) < ( x − 3)(2 x +6) E.9821 Solve the following inequalities: a ( x − 2)( x + 1) > (2 x + 1)(2 x + 2) b (3 x − 4)(5 − 2 x ) (4 x − 10)(2 − 3 x ) E.9823 Solve the inequation : (3 x − 2)(4 − 2 x ) > 2(3 − 2 x )( x − 2) 12. Remarkable inequalities and identities E.483 Solve the inequation : 16 x 2 25( x + 1) 2 E.9789 Solve the inequation : x 2 − 1 x + 2 < 0 E.8191 Solve the inequation : 2 x − 1 x 2 + 6 x + 9 < 0 E.9826 Solve the inequation : ( x +1) 2 x 2 − 1 E.482 Solve the following inequalities in R : a 9 x 2 + 36 x + 36 2 x − 3 < 0 b x 2 − 5 3 x 2 + 2 3 x + 1 0 E.456 1 Find the factorization : 2 x 2 + 2 6 x + 3 = ( 2 x + 3) 2 2 Solve the inequality: 2 x 2 + 2 6 x + 3 (4 x − 1)(3 − x ) < 0 E.9822 Solve the following inequalities: a (4 x + 4)( x + 2) < − 1 b x (4 x − 3) > ( x − 1)(3 x + 4) E.461 Solve the following inequalities and give the set of solutions in interval form. a x 2 + x + 1 ( x + 1)( x − 1) b x + 2 2 <x 2 + 5 x − 2 13. Inequation and rational fractions E.9817 a Study the sign on R of the expression : x + 1 x − 1 + 1 b Solve the inequation : x + 1 x − 1 < − 1 E.447 Solve the following inequalities: a 1 1 + x < 1 1 − x b 1 x + 1 x + 5 0 E.475 Solve the inequation : 3 x + 1 − 2 x + 1 > 1 E.2856 1 Determine the expression for P to perform the following factorization : 2 x 2 + x − 1 = ( x + 1) × P 2 Draw up the sign table for : 2 x 2 + x − 1 x 2 − 4 3 Solve the following inequation : 5 x 2 + x − 13 x 2 − 4 3 E.4816 Solve the following inequalities: a 1 x 1 5 b 1 x > 3 4 c 1 x < − 1 3 d 1 x < 2 https://chingmath.fr chapExoCorrec/473 sacados/473 chapExoCorrec/6688 sacados/6688 chapExoCorrec/9791 sacados/9791 chapExoCorrec/6686 sacados/6686 chapExoCorrec/9807 sacados/9807 chapExoCorrec/9821 sacados/9821 chapExoCorrec/9823 sacados/9823 chapExoCorrec/483 sacados/483 chapExoCorrec/9789 sacados/9789 chapExoCorrec/8191 sacados/8191 chapExoCorrec/9826 sacados/9826 chapExoCorrec/482 sacados/482 chapExoCorrec/456 sacados/456 chapExoCorrec/9822 sacados/9822 chapExoCorrec/461 sacados/461 chapExoCorrec/9817 sacados/9817 chapExoCorrec/447 sacados/447 chapExoCorrec/475 sacados/475 chapExoCorrec/2856 sacados/2856 chapExoCorrec/4816 sacados/4816
-2-1234I2JOCf E.457 Consider the following algebraic ex-pression : 2 x + 7 x + 3 − 4 x + 4 2 x + 1 1 Reduce the previous expression to the same denomina-tor. 2 Draw up the sign table for this expression. 3 Solve the inequation : 2 x + 7 x + 3 4 x + 4 2 x + 1 E.2876 Solve the following inequalities: a 2 x − 4 4 x + 1 3 x + 5 6 x b 4(2 x + 1) 4 x − 1 + 2 − 4 x 2 x + 3 0 E.4916 Solve the inequation : 6 − 2 x 2 x + 4 x − 3 5 − x E.9825 Solve the inequation : 5 x + 1 2 x − 1 + 3 x + 3 x + 1 0 E.9818 1 Establish equality: 3 x − 6 2 x + 3 − 4 − 7 x 2 x − 2 = 5 x (4 x − 1) (2 x − 2)(2 x + 3) 2 Solve the inequation : 3 x − 6 2 x + 3 < 4 − 7 x 2 x − 2 E.2860 1 Expand the following expression : ( x − 1)( x +3) 2 Solve the inequation : 5 x + 1 1 − 2 x + 3 x + 3 x > 0 E.9824 Solve the following inequations : x 2 − x 2 x + 4 0 14. Algebraic manipulations E.300 1 Here are four inequalities solved by students. Each one contains one or more errors. Identify the error(s) made by each student : Student 1: 5 x + 2 7 x − 3 12 x − 1 x − 1 12 S = −∞ ; 1 2 Student 2: 3 x − 8 4 x + 2 + 2 − x 10 + 2 x − 10 − 2 S = −∞ ; − 10 − 2 Student 3: 1 x < 1 1 < 1 × x 1 <x S = 1 ; + ∞ Student 4: x 2 x x 1 S = 1 ; + ∞ 2 Correctly restate the solution to each of these inequali-ties. E.299 Consider two positive numbers a and b such that a b . For c ∈ R , we wish to compare the numbers a · c and b · c . 1 a Give the sign of a − b . b Depending on the sign of c , determine the sign of ( a − b ) × c . 2 For a , b , c three real numbers, complete the following statements using comparison signs : Si a b et c 0 Alors a · c . . . . . . b · c Si a b et c 0 a · c . . . . . . b · c E.341 In this exercise, a and b denote posi-tive real numbers 1 For a b , prove the following two inequalities: a 2 a × b ; b 2 a × b Complete the sentence : If 0 a b , then . . . 2 Suppose that a 2 b 2 : a Factor the expression a 2 − b 2 b Determine the sign of a − b . Compare a and b . c Complete the following sentence : If a and b are positive, and a 2 b 2 , then : : : 3 If a and b are any two numbers such that a 2 b 2 , can we deduce a comparison between the numbers a and b ? E.333 Let x and y be two real numbers with x =0 , show that y x 2 and y +3 x 2 +2 are arranged in the same direction as 2 · y and 3 · x 2 . 15. Graphical solutions of inequalities E.11488 Consider the function f defined on R , whose curve C f is given in the coordinate plane : https://chingmath.fr chapExoCorrec/457 sacados/457 chapExoCorrec/2876 sacados/2876 chapExoCorrec/4916 sacados/4916 chapExoCorrec/9825 sacados/9825 chapExoCorrec/9818 sacados/9818 chapExoCorrec/2860 sacados/2860 chapExoCorrec/9824 sacados/9824 chapExoCorrec/300 sacados/300 chapExoCorrec/299 sacados/299 chapExoCorrec/341 sacados/341 chapExoCorrec/333 sacados/333 chapExoCorrec/11488 sacados/11488 -2-1234I2JOCf
-6-5-4-3-2-12345678I-123JOCg -7-6-5-4-3-2-12345I-4-3-2-123JOCf -12345678I-12345JO -5-4-3-2-12345I-3-2-123JOCfCf Graphically, give the set of solutions to the inequalities: a f ( x ) 1 b f ( x ) 0 ; 75 E.11607 In an orthonormal coordinate sys-tem ( O ; I ; J ) , consider the curve C g representing the function g : 1 Give, without justification, the domain of the function g . 2 Give, without justification, the solutions to the following two equations : a g ( x ) = 2 b g ( x ) = 1 3 Solve the inequalities graphically: a g ( x ) 2 b g ( x ) < 1 Highlight the parts of the curve C g used to answer these questions. E.380 In a coordinate system O ; I ; J , con-sider the curve C f representing the function f : Answer the following questions graphically: 1 Give the domain of this function. 2 Graphically, solve the inequalities: a f ( x ) 1 b f ( x ) − 1 E.2752 In an orthonormal frame of refer-ence ( O ; I ; J ) , consider the function f defined on the interval 0 ; 7 whose graphical representation is given below : Graphically solve the following inequalities: a f ( x ) 1.5 b f ( x ) 1 We’ll leave a few construction lines. E.4384 In the reference frame ( O ; I ; J ) be-low is represented the curve C f representative of the function f Graphically solve the following two inequalities: a f ( x ) 1.5 b f ( x ) − 1 https://chingmath.fr chapExoCorrec/11607 sacados/11607 -6-5-4-3-2-12345678I-123JOCg chapExoCorrec/380 sacados/380 -7-6-5-4-3-2-12345I-4-3-2-123JOCf chapExoCorrec/2752 sacados/2752 -12345678I-12345JO chapExoCorrec/4384 sacados/4384 -5-4-3-2-12345I-3-2-123JOCfCf
-7-6-5-4-3-2-123I-2-123JOCfCf -7-6-5-4-3-2-123I-2-123JOCfCf -6-5-4-3-2-123456I-4-3-2-1234JOCfCf E.2782 Consider the function f whose repre-sentation is given below in the frame ( O ; I ; J ) orthonormal : All questions will be answered using the graph below. 1 Give the definition set of the function f . 2 The construction features necessary to answer the follow-ing questions will be left : a Determine the image of − 5.5 by the function f . b Determine the set of antecedents of 1 by the function f . 3 By highlighting the relevant parts of the curve C f , graph-ically solve the following two inequations : a f ( x ) − 1 b f ( x ) 0 E.4914 Consider the function f whose rep-resentation is given below in the reference frame ( O ; I ; J ) orthonormal : All questions will be answered using the graph below. 1 Give the definition set of the function f . 2 Answers to the following questions will be justified : a Determine the image of the number 1 by the function f . b Determine the set of antecedents of the number 1.5 by the function f . 3 Determine the image of the following intervals by the function f : a − 5.5 ; − 2 b 0 ; 2 4 Determine, for each question, the set of real x verifying the following inequalities: a f ( x ) 0 b 1 <f ( x ) 2 E.532 The plane is given the reference frame ( O ; I ; J ) orthonormal. Below is the curve C f representative of the function f : 1 Give the definition set of the function f . 2 Determine, graphically, the image of the following num-bers by the function f : a − 3 b 1 c 2 3 Determine, graphically, the set of antecedents of the num-ber 2 by the function f . 4 a Graphically solve the inequation : f ( x ) 2 . b Graphically solve the inequation : f ( x ) < 0 . 16. Unclassified exercises E.858 1 a Solve the following inequation : 7 x − 2 > 3 x + 6 b On a graduated line, hatch the set of solutions to this inequation. 2 Solve the equation : 3(5 x − 7)( x − 2) = 0 E.2330 Consider the following numbers : A = 1001 × 999 − 999 2 ; B = 57 × 55 − 55 2 C = ( − 2) × ( − 4) − ( − 4) 2 1 a Give the values read on the calculator for A , B and C . b Are the integers A and B prime with each other? Jus-tify briefly. 2 We pose : D =( x +1)( x − 1) − ( x − 1) 2 a x being an integer, greater than 1, show that D is a https://chingmath.fr chapExoCorrec/2782 sacados/2782 -7-6-5-4-3-2-123I-2-123JOCfCf chapExoCorrec/4914 sacados/4914 -7-6-5-4-3-2-123I-2-123JOCfCf chapExoCorrec/532 sacados/532 -6-5-4-3-2-123456I-4-3-2-1234JOCfCf chapExoCorrec/858 sacados/858 Antilles-Guyane - Septembre 2004 chapExoCorrec/2330 sacados/2330 France - 2008
ABCDEF2x−3x 5cm‘ABCD multiple of 2. b For what values of x , is D a negative or zero number? Plot the values found on an axis, hatching the part that doesn’t fit. 3 Find an expression E of the same form as A for which the result of the calculation is 2008. E.2511 In each row of the table, three statements are proposed. Only one is correct. For each line, copy the number of the correct proposition on the copy: Proposal 1 Proposal 2 Proposal 3 2 5 + 5 12 − 1 15 = 23 30 2 5 + 5 12 − 1 15 = 3 2 5 + 5 12 − 1 15 =0.75 8 25 ÷ 16 75 = 2 3 8 25 ÷ 16 75 = 3 2 8 25 ÷ 16 75 = 1 6 16 + 9 = 7 16 + 9 = 5 16 + 9 = 12 (2 x − 5) 2 = 4 x 2 − 14 x + 25 (2 x − 5) 2 = 4 x 2 − 20 x + 25 (2 x − 5) 2 = 4 x 2 − 25 49 x 2 − 25 = (7 x − 5) 2 49 x 2 − 25 = (7 x − 5)(7 x +5) 49 x 2 − 25 = (7 x − 5)(7 x − 5) ( − 2) is a solution of the equation : ( x − 2)(2 x +4)=0 ( − 2) is a solution of the equation : x 2 + 4 = 0 ( − 2) is a solution of the equation : − 2 x + 4 = 0 102 is a solution of the inequation 2 x + 1 3 102 is a solution of the inequation − 2 x + 1 3 102 is a solution of the inequation − 2 x + 1 > 3 E.854 india, June 1999 1 Solve the inequation : 2 x − 3 x +1 2 x denoting a number greater than or equal to 4, ABCD is a square whose side measures 2 x − 3 . a Show that the area of the rectangle BCEF is ex-pressed by the formula : A ( x ) = (2 x − 3) 2 − (2 x − 3)( x +1) b Expand and reduce A ( x ) . c Factorize A ( x ) . d Solve the equation : (2 x − 3)( x − 4)=0 e For what value(s) of x is the area of BCEF zero? E.6430 Consider a rectangle ABCD with a length of 5 m and a width ‘ whose value is unknown, but whose boundaries are known as follows : 2 ‘ 3 1 Give a range for the perimeter P of the rectangle ABCD . 2 Give a range for the area A of the rectangle ABCD . E.466 Solve the following inequalities: a ( x + 1) 2 > 0 b ( x + 1) 2 0 c ( x + 1) 2 < 0 d x 2 + 1 0 e x 2 − 4 < ( x + 2) 2 f ( x + 1) 2 − ( x − 1) 2 0 E.306 Let n be a relative integer ( n ∈ Z ) . Determine the set of solutions of equation : − 3 · n 2 + 5 > − 13 E.343 1 For each question, plot the set of numbers verifying the frame on a graduated line: a − 1 x 2 b 2 x < 3 c x > 9 d − 5 > x 2 Write down in interval form the sets represented previ-ously on a graduated line. https://chingmath.fr chapExoCorrec/2511 sacados/2511 Clermont-Ferrand - Septembre 2001 chapExoCorrec/854 sacados/854 Inde - juin 1999 ABCDEF2x−3x chapExoCorrec/6430 sacados/6430 5cm‘ABCD chapExoCorrec/466 sacados/466 chapExoCorrec/306 sacados/306 chapExoCorrec/343 sacados/343