Model Test – Solved ഉത്തരങ്ങൾ

−7 is 7 units from 0, so |−7| = 7.

Bigger minus smaller: 5 − (−3) = 8. (Or |−3 − 5| = |−8| = 8.)

½(−6 + 2) = ½ × (−4) = −2.

−3/4 is a fraction; √2, π and √3 cannot be written as fractions.

Both 4 and −4 are 4 units from 0.

−2 lies to the right of −5 on the number line, so −2 > −5.

|x − y| is the distance between x and y (the bigger minus the smaller).

A number and its negative are equally far from 0.

x is 6 units from 4: to the right 4 + 6 = 10; to the left 4 − 6 = −2. (Algebra: x − 4 = 6 or x − 4 = −6.)

Write x + 3 = x − (−3): x is 2 units from −3, so x = −3 + 2 = −1 or x = −3 − 2 = −5.

x is less than 3 units from −2: −5 < x < 1. The integers strictly between −5 and 1 are −4, −3, −2, −1, 0.

(i) ½(−3½ + 1½) = ½ × (−2) = −1. (ii) ½(−3 + 7) = ½ × 4 = 2.

(i) 4½ − (−6¼) = 4½ + 6¼ = 10¾. (ii) −0.7 is bigger: −0.7 − (−2.5) = 1.8.

Multiplying never changes the size of the answer, only its sign: the size of xy is the size of x times the size of y. If x, y have the same sign, xy is positive and |xy| = xy = |x||y|; if they have opposite signs, xy is negative and |xy| = −xy = |x||y|; if either is 0, both sides are 0. (Or: |xy|² = (xy)² = x²y² = |x|²|y|², and both sides are positive.)

(i) |6 + (−2)| = |4| = 4 while |6| + |−2| = 8: smaller, because the numbers pull in opposite directions. (ii) |−6 + (−2)| = |−8| = 8 = 6 + 2. In general |x + y| ≤ |x| + |y|, with equality exactly when x and y have the same sign or one of them is 0.

(i) x is equally far from 2 and 8, so it is their midpoint: ½(2 + 8) = 5. (ii) |x + 1| = |x − (−1)|: x is equally far from −1 and 5, so x = ½(−1 + 5) = 2.

The midpoint of −2 and 4 is ½(−2 + 4) = 1. The midpoint of −2 and 1 is ½(−2 + 1) = −½; of 1 and 4 is ½(1 + 4) = 2½. Check: the four parts are each 1½ long (the whole is 6).

(i) |x| is the distance between 0 and x on the number line: |x| = x if x > 0, |x| = −x if x < 0, |x| = 0 if x = 0. (ii) x is less than 3 away from 1: between 1 − 3 = −2 and 1 + 3 = 4, so −2 < x < 4 (an open interval). (iii) x is more than 5 away from 3: to the left of 3 − 5 = −2 or to the right of 3 + 5 = 8 – two rays. (iv) The integers strictly between −2 and 4: −1, 0, 1, 2, 3.

(i) ½(1 + 2) = 1½. (ii) midpoint of 1 and 1½ is 1¼; of 1½ and 2 is 1¾. (iii) The midpoint of any two numbers lies between them, so between x and y there is ½(x + y); between x and that midpoint there is another, and so on – the process never stops, so there are infinitely many numbers between them (density, സംഖ്യാസാന്ദ്രത). (iv) ½(−½ + (−⅓)) = −5/12; ½(−½ + (−5/12)) = −11/24; ½(−5/12 + (−⅓)) = −3/8. Order: −½ < −11/24 < −5/12 < −3/8 < −⅓.