Grade 8 / literal expressions: relative numbers 45 exercises (100% corrected)

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ChingQuizz : 12 exercises available for Quizz assessment : 1. Reminders on simplification E.1081 Reduce, if possible, the following literal expressions : a 12 + 5 x b 4 × 5 x c 3 x − ( − 3) × x d − 7 x + 9 x 2 e 2 x 2 + 7 x 2 f 9 x 2 + 5 x − 4 − 2 x + 5 x 2 E.4521 Expand, then reduce the following expressions : a 3 × (2 x + 1) − 3 x × 5 b 2( x − 1) + x (2 − x ) c − 5 × 3 x + 3 × (5 − x ) d x (2 − x ) + 3(3 + x ) E.7929 Factor the following expressions : a 3 x + 3 b 6 x + 9 c 3 x 2 + 2 x d x 2 + x 2. The opposite of a number E.1715 1 a What are the numbers − 4 and 4 called, between them? b What are the numbers 3 and 1 3 called, between them? Definition: Two numbers a and b are said to opposés if their sums are 0 Two numbers a and b are said inverses if their product is 1 . Note: two numbers are opposite ’they have the same distance to zero but are of opposite signs. two rational numbers are inverse if their numerator and denominator are reversed between each other. 2 Complete, line by line, the table below : a b a + b a − b − a + b − a − b 2 3 − 4 5 9 − 2 3 Using the previous table as a guide, complete the follow-ing sentences : For any number a and b , the opposite of ( a + b ) is : − ( a + b ) = : : : : : : : : : : : : For any number a and b , the opposite of ( a − b ) is : − ( a − b ) = : : : : : : : : : : : : E.1074 Proposition: the opposite of a sum is equal to the sum of the opposites. Examples: − (5+4)= − 5 − 4 − (7 − 3)= − 7+3 For each question, name the number ˇ intruder ı : a b c 1 − (5 + 1) − 5 + 1 − 5 − 1 2 7 − 3 − (7 − 3) − 7 + 3 3 − ( − 2.5 + 3) 2.5 − 3 2.5 + 3 4 − ( x − 2) − x − 2 − x + 2 E.1860 For each question, give the intruder value : a b c d 1 − (2 × 3) ( − 2) × 3 2 × ( − 3) ( − 2) × ( − 3) 2 − − 4 × 5 4 × ( − 5) − 4 × ( − 5) 4 × 5 3 − (2 × 3 × 4) ( − 2) × 3 × 4 2 × 3 × ( − 4) − 2 × ( − 3) × 4 Note: to take the opposite of a product, we change only one of the factors to its opposite E.1918 Complete the following dotted lines to verify equality: a − (3 + 4) = : : : 3 : : : 4 b − ( − 2 + 1) = : : : 2 : : : 1 c − ( − 4 − 7) = : : : 4 : : : 7 d − (7 − 9) = : : : 7 : : : 9 E.7923 Complete the following blanks to verify equality: a − ( − 4 + 12) = : : : 4 : : : 12 b − (3 + 7) = : : : 3 : : : 7 https://chingmath.fr chapExoCorrec/1081 sacados/1081 chapExoCorrec/4521 sacados/4521 chapExoCorrec/7929 sacados/7929 chapExoCorrec/1715 sacados/1715 chapExoCorrec/1074 sacados/1074 chapExoCorrec/1860 sacados/1860 chapExoCorrec/1918 sacados/1918 chapExoCorrec/7923 sacados/7923
3x33x−3−3x−3−3x3−(5x2(4x−(4x−−(3−xx(2x−2(1−x2−5−(5−2xx(1−x8x2 E.1736 For each question, look for the odd one out : a b c 1 − (2 × 5 + 3) − 2 × 5 + 3 − 2 × 5 − 3 2 − (3 − 5 × 2) − 3 + 5 × 2 − 3 − 5 × 2 3 − (2 × 3+4 × 5) ( − 2) × 3+4 × 5 − 2 × 3 − 4 × 5 E.1076 Which of the 4 expressions below is the odd one out? a − 3 × (2 x − 3) b ( − 3) × (2 x − 3) c 3 × ( − 2 x + 3) d ( − 3) × ( − 2 x + 3) E.7920 Which of the 4 expressions below is the odd one out? a − 2 × (4 x + 1) b ( − 2) × (4 x + 1) c 2 × ( − 4 x − 1) d ( − 2) × ( − 4 x − 1) 3. Removal of parentheses in a sum or difference E.1737 Give the reduced form of each of the following expressions : a − (2 x + 1) b 3 − (5 − x ) c 2 − (2 x − 1) d 3 x − ( − 2 x − 1) E.1919 Give the contracted form of each of the following expressions : a 3 − (2 + x ) + 5 x b x 2 + 2 − (2 × x + 1) E.1923 Give the reduced form of each of the following expressions : a 3 − (2 x + 1 − x 2 ) b ( x + 1) − (2 − x ) E.4475 Give the reduced form of each of the following expressions : a (3 x + 4) − ( x 2 − 4 x + 2) b − ( x + 3) + x 2 − x + 2 c − ( x 2 − 2) + (3 x 2 + 4 x ) E.8974 Give the reduced form of each of the following expressions : a ( x 2 + 3 x + 4) − (5 x 2 + 6 x + 7) b − (2 x − 5 x + 1 − 4 + 7 x ) c (3 x + 2) − 5 x + 6 − ( − 6 x + 2) E.8972 Give the reduced form of each of the following expressions : a 3 x − ( x 2 + 4) − 5 x + 5 b − ( x − 2) + (3 − x ) + 5 x E.8970 Give the reduced form of each of the following expressions : a 2 x 2 + 5 − (2 x − 5) b 5 − (2 x − 4) − 2 x 2 + x E.8973 Give the reduced form of each of the following expressions : a − ( − 3 x − 1) − (5 x − 2) b − (3 x 2 + 4 x − 8) − (2 x − 4) E.7935 Reduce the following expressions : a − 3 x 2 + 5 x − 9 − − 5 x + 1 b − x − 2 + 2 x + 2 − 3 − x 2 E.1946 Give the reduced form of each of the following expressions : a − (2 x + x 2 − 4) − ( − 2 x 2 + 4 x − 3) b − (2 x 2 − 2) + ( x − 2) − (2 − x ) c 3 x − (2 x 2 + 1 − 8 x ) − (3 x − 2 x 2 + 7) E.8971 Give the reduced form of each of the following expressions : a +( x + 1) − (3 x 2 + 2) + 3 − (2 + x ) b 2 − 3 − ( x − 2) c − 2 − (1 − x ) + 1 − (3 − 2 x ) E.4598 Copy and complete the table be-low, linking the expressions in the left-hand table with their reduced expanded forms obtained in the right-hand table. 4. Parenthesis and simple distributivity E.1075 Expand and then simplify the fol-lowing expressions : a − 3 × (2 − x ) b − 2( x + 2) c − 2 x ( x − 2) d − 2 × (3 x − 1) https://chingmath.fr chapExoCorrec/1736 sacados/1736 chapExoCorrec/1076 sacados/1076 chapExoCorrec/7920 sacados/7920 chapExoCorrec/1737 sacados/1737 chapExoCorrec/1919 sacados/1919 chapExoCorrec/1923 sacados/1923 chapExoCorrec/4475 sacados/4475 chapExoCorrec/8974 sacados/8974 chapExoCorrec/8972 sacados/8972 chapExoCorrec/8970 sacados/8970 chapExoCorrec/8973 sacados/8973 chapExoCorrec/7935 sacados/7935 chapExoCorrec/1946 sacados/1946 chapExoCorrec/8971 sacados/8971 chapExoCorrec/4598 sacados/4598 3x33x−3−3x−3−3x3−(5x2(4x−(4x−−(3−xx(2x−2(1−x2−5−(5−2xx(1−x8x2 chapExoCorrec/1075 sacados/1075
E.9006 Expand, then reduce the following expressions : a − 2 x + 5 b − x x + 2 E.1929 Expand and reduce the following expressions : a 3 − 2 × (2 x − 1) + x b 7 × ( x + 2) − 3 × (1 + x ) c 2 × ( x − 1) − 4( x − 3) d − 2 × ( x − 1) + 4(2 x − 3) E.8968 Expand, then reduce the following expressions : a 3 − 4 × (3 − x ) b 3 × (4 x − 2) − 2 × (3 − x ) E.9007 Expand, then reduce the following expressions : a 5 − 2 2 x + 4 b 2 x + 1 − 3 3 − x E.8969 Expand, then reduce the following expressions : a 3( x − 2) − 3(2 x − 1) b ( − 2 + 3 x )( − 2) − (6 x + 2) c 3(5 − 2 x ) − 2(3 + 2 x ) d ( x − 2) × 2 − (2 x + 5) E.4549 Expand, then reduce the following expressions : a 3 x ( x +2) − ( − 3 x 2 − 4 x ) b 2(3 − x +2) − 2 x (3 − 2 x ) E.1082 Expand and reduce the following expressions : a 3 x (2 x − 4) − 5(4 − x ) b − ( x + 2) + 3(2 x 2 + 1) E.8976 Expand, then reduce the following expressions : a 4 x (3 x − 4) + 2( x 2 + 2) b ( x 2 + 4 x + 3) − (3 − x 2 ) E.4474 Expand, then reduce the following expressions : a 3( x 2 +4 x +1)+2( x 2 − 1) b 5 x − 2 − ( x 2 + 2) c 2( − x 2 +5 x +4+ x ) − (3 x − 7) d − 2 x ( x +1) − 2(3 x − 5) E.8967 Expand, then reduce the following expressions : a x × (2 − x ) − 3 × ( x 2 − 1) b 2 − ( x + 1) × x c − (2+ x ) × 3 + x × ( − x +1) d − x × (2 x − 4) − (3 − 2 x 2 ) E.8975 Expand, then reduce the following expressions : a − 2 3 − 7 × (2 − x ) b − 2 2 − (1 − x ) c 1 − 2 x 1 − ( x + 1) 5. Equality of literal expressions E.1927 Method definition : To show that two expressions are different , , we show that for a value of x , their evaluation gives distinct values. This number is then called a counterexample to the equality . Establish that the following equalities are false : a ( x + 1)(2 x − 1) = x 2 + x b 3 − ( x 2 + x ) = (3 x + 1)(3 − x ) c x 2 + x + 4 = (5 x + 1)(4 − 5 x ) E.7927 Definition: Two expressions are equal if they have the same value, regardless of the value of x (This is referred to as identity) . Method : To establish an identity, we expand and sim-plify each expression to show that they have the same ex-panded simplified expression . Establish that the following identities are true : a (2 x − 1)(1 − x ) = − 2 x 2 + 3 x − 1 b (3 x + 1)(2 x − 2) = 6 x 2 − 2(2 x + 1) E.4600 Consider the following two expres-sions : A = 3 x 2 + 5 x + 1 ; B = (2 x + 1) 2 Justify that the expressions A and B are not equal. E.4490 Consider the two expressions below : A = 2 x 2 + x − 7 ; B = 3( x − 2) + 3 1 a Evaluate the expressions A and B for x =2 . b Evaluate expressions A and B for x = − 1 . 2 Are the two expressions A and B equal for all values of x ? Justify your assertion. E.7928 Consider the following two expres-sions : C = (3 x − 2)(1 + 2 x ) ; D = x × (6 x − 1) − 2 Using expansions and simplifications, show that the expres-sions C and D are equal. https://chingmath.fr chapExoCorrec/9006 sacados/9006 chapExoCorrec/1929 sacados/1929 chapExoCorrec/8968 sacados/8968 chapExoCorrec/9007 sacados/9007 chapExoCorrec/8969 sacados/8969 chapExoCorrec/4549 sacados/4549 chapExoCorrec/1082 sacados/1082 chapExoCorrec/8976 sacados/8976 chapExoCorrec/4474 sacados/4474 chapExoCorrec/8967 sacados/8967 chapExoCorrec/8975 sacados/8975 chapExoCorrec/1927 sacados/1927 chapExoCorrec/7927 sacados/7927 chapExoCorrec/4600 sacados/4600 chapExoCorrec/4490 sacados/4490 chapExoCorrec/7928 sacados/7928
ABCDIJKLMNOPQRSTx5cm 2x3x4× 2x3x42x28x3x122x211x12F.O.I.L. 2x1x4×Tableau 1 2x2x8x42x1x4×Tableau 1 abcda×ca×db×cb×dproductoftheFirsttermesproductoftheOutertermesproductoftheInnertermesproductoftheLasttermesFOIL 2x3x510x152x23x13x152x2× 2x−32x2−4x36x−9−8x212x4x3−6x2×4x3−14x218x−9 E.4601 Consider the figure op-posite : où the quadrilaterals ABCD , AIJK , BRST , COPQ and DLMN are squares. Information on the lengths is given in the figure. The polygon IJKLMNOPQRST is designated P . Part A 1 We’re interested in the special case where x =2 . a Determine the areas of squares ABCD and DLMN . b Deduct the area of the polygon P for the value x =2 . 2 Justify that the area A of the polygon P is expressed as a function of x by: A =25 − 4 x 2 Part B 3 We’re interested in the special case where x =1 . a Determine the areas of the rectangles KLQR and MPSJ . b Deduct the area of the polygon P for the value x =1 . 4 Justify that the area of the polygon P is expressed as a function of x by: A =10(5 − 2 x ) − (5 − 2 x )(5 − 2 x ) Part C 5 Justify that the expressions obtained in questions 2 and 4 are equal. 6. Usmath E.9647 With American notations: to perform double distribu-tivity, we can use a table to represent each of the prod-ucts of the F.O.I.L. For example, to expand the expression 2 x +3 x +6 , we use the table : We obtain the following development : 1 a Complete the table 1 b Deduce the reduced expansion form of the expression : A = 2 x + 1 x + 4 2 a Complete the table 2 b Deduce the reduced expansion form of the expression : B = x − 3 3 x − 1 E.9646 With American notation: double distributivity is associ-ated with the acronym F . O . I . L . whose four letters represent the four products : Complete the table below to obtain the four terms obtained by double distributivity: Product of the F . terms O . terms I . terms L . terms 3 x + 2 x + 4 x − 2 2 x + 1 3 − x 5 x − 2 E.9652 With American notation: Algebraic expressions are sometimes multiplied in columns, as shown below : Write your operations in a row and perform the following multiplications: a 2 x + 1 3 x − 2 b 3 x 2 − 5 x − 1 3 x − 2 c 5 − 2 x x 2 + 1 d x + 1 − 3 x 2 + 2 x − 1 7. Unclassified exercises https://chingmath.fr chapExoCorrec/4601 sacados/4601 ABCDIJKLMNOPQRSTx5cm chapExoCorrec/9647 sacados/9647 2x3x4× 2x3x42x28x3x122x211x12F.O.I.L. 2x1x4×Tableau 1 2x2x8x42x1x4×Tableau 1 chapExoCorrec/9646 sacados/9646 abcda×ca×db×cb×dproductoftheFirsttermesproductoftheOutertermesproductoftheInnertermesproductoftheLasttermesFOIL chapExoCorrec/9652 sacados/9652 2x3x510x152x23x13x152x2× 2x−32x2−4x36x−9−8x212x4x3−6x2×4x3−14x218x−9
E.8924 Expand and reduce the following expression : ( − 3 x ) × (2 − x ) + 3 × ( x 2 + 3) https://chingmath.fr chapExoCorrec/8924 sacados/8924