Solve the following inequations in R.

1. 1/x-1≤2

2. 5x+8/4-x<2

3. x/x-5>1/2

4. 0<-x/2<3

5. |3x-4/2|≤5/12

6. |x-2|/x-2>0

7. |x+2|-x/x<2

8. |2x-1/x-1|>2

9. |x-1|+|x-2|+|x-3|≥6

10. The marks scored by rohit in two tests were 65 and 70. Find the minimum marks he should score in third test to have an average of at least 65 marks.

11. The longest side of a triangle is three times the shortest side and the third side is 2 cm shorter than the longest side if the perimeter of the triangle at least 61cm, find the minimum length of the shortest-side.

12. How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

13. A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If there are 640 litres of the 8% solution, how many litres of 2% solution will have to be added?

14. The water acidity in a pool is considered normal when he average pH reading of three daily measurements is between 7.2 and 7.8. If the first two pH reading are 7.48 and 7.85, find the range of pH value for the third reading that will result in the acidity level being normal.

15. Show that the solution set of the following linear inequations is empty set:

15.1) x-2y ≥0, 2x-y ≤-2,x ≥0, y ≥0

15.2) x+2y≤3, 3x+4y ≥12, y ≥1, x ≥0, y ≥0

13)

Let the required quantity of 2% boric acid = *x *liters

So, quantity of mixture = 640 + *x* liters

The resulting mixture is to be more than 4% but less than 6% boric acid.

So, by the given condition

8% of 640 + 2% of *x* > 4% of (640 + *x*) …… (1)

And

8% of 640 + 2% of *x* < 6% of (640 + *x*) …… (2)

Equation (1) gives,

(8/100)640 + (2/100) *x* > (4/100)(640 + *x*)

5120+2* x* > 2560+4* x*

2560 > 2* x*

*x *< 1280 …… (3)

Equation (2) gives,

(8/100)640 + (2/100)* x* < (6/100)(640 +* x*)

5120 + 2* x* < 3840 + 4x

1280 < 4* x*

*x *> 320 …… (4)

From equation (3) and (4),

320 < x < 1280

Hence, quantity of 2% boric acid lies between 320 liters and 1280 liters.

15)

**Given,** x- 2y ≥ 0, 2x - y ≤ -2, x ≥ 0, y ≥ 0

On converting the given inequations into equations, we get

x- 2y = 0, 2x - y = -2, x = 0, y = 0

Now, consider the line x - 2y = 0. Its solution set is:

x | 0 | 2 |

y | 0 | 1 |

We find that (0,0) satisfies the inequation x- 2y ≥ 0. So, the portion containing the origin represents the solution set of the inequation x- 2y ≥ 0.

Again, consider the line 2x - y = -2. Its solution set is:

x | 0 | -1 |

y | 2 | 0 |

We find that (0,0) doesn't satisfy the inequation 2x - y ≤ -2. So, the portion not containing the origin represents the solution set of the inequation 2x - y ≤ -2.

Clearly, x ≥ 0 and y ≥ 0 represents the first quadrant.

As all the four lines doesn't possess any common region. So, the solution set of the given linear inequations is empty.

14)

Let *x *be the third reading of the pH level.

The acidity of the pool is normal when the average pH reading measurements is between 7.2 and 7.8

The first two measurements are 7.48 and 7.85

The average

Similarly,

The average

Thus the range third reading of the acidity of the pool should be .

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