# F.4 math---inequalities3

(1)

Mr. Lee drove from town A to town B at 65km/h. Half an hour later, Mr. Ho drove along the same route from town A to town B at 80km/h. At least how long would it take for Mr. Ho to be not less than 65km ahead of Mr. Lee?

(2)

Ivan and Tony have \$x and \$(x+50) respectively. It is known that half of the amount with Tony is at least \$10 more than the amount with Ivan, and the sum of the amounts with them is greater than \$60.

(a) Find the range of the amount with Ivan.

(b) Find the range of the amount with Tony.

Rating

(1) Let t be the number of hours Mr Ho started off from town A

The distance of Mr Lee with respect to the departure time of Mr Ho is (t + 0.5)*65 km = 65t + 32.5 km

The distance of Mr Ho is 80t km

In order for Mr Ho to be not less than 65 km ahead of Mr Lee,

80t - (65t + 32.5) >= 65

15t - 32.5 >= 65

15t >= 97.5

t >= 6.5 hours

It takes 6.5 hours for Mr Ho to be ahead of Mr Lee by at least 65 km

(2) (x + 50)/2 - x >= 10 ...(1)

x + (x + 50) >= 60 ... (2)

(1) => x + 50 - 2x > = 20

-x >= -30

x <= 30

(2) => 2x + 50 >= 60

2x >= 10

x >= 5

(a) The range of the amount with Ivan is \$5 <= x <= \$30

(b) The range of the amount with Tony is 5 + 50 <= x+50 <= 30 + 50

or \$55 <= x + 50 <= \$80

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