Linear Inequalities Miscellaneous Exercise
Question:
Solve the inequalities −12<4+3x/5≤2
Answer:

−12<4+3x/5≤2

12 < 4+3x/5 and 4+3x/5 ≤ 2

12 < 4+3x/5 and 4+3x ≤ 10

8 < 3x/5 and 3x ≤ 10

5.33 < x and x ≤ 10
Question:
Solve the inequalities −3≤4−7x/2≤18
Answer:

3 ≤ 4  7x/2 ≤ 18

3 ≤ 4  (7/2)x ≤ 18

3 ≤ (8/2)  (7/2)x ≤ 18

3 ≤ 1  x ≤ 18

x ≥ 2 and x ≤ 19
Question:
Solve the inequalities 6≤−3(2x−4)<12
Answer:
 6≤−3(2x−4)<12
 6≤−6x+12<12
 −6x<6<12
 −6x<12
 x>−2
Question:
Solve the inequalities and represent the solution graphically on number line. 3x−7>2(x−6),6−x>11−2x
Answer:
 3x  7 > 2(x  6)
3x  7 > 2x  12
3x  2x > 12  7
x > 5
 6  x > 11  2x
6  x > x  11
7  x > 11
x < 11
The solution is x < 11 and x > 5.
Graphically:
—————————— 5 6 7 8 9 10 11
Question:
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 we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?
Answer:

Calculate the amount of boric acid in the 8% solution: 8% of 640 litres = 51.2 litres

Calculate the amount of boric acid in the desired mixture: 4% of 640 litres = 25.6 litres

Calculate the amount of boric acid in the 2% solution that needs to be added to the 8% solution: 25.6 litres  51.2 litres = 25.6 litres

Calculate the amount of 2% solution that needs to be added to the 8% solution: 25.6 litres / 0.02 = 1280 litres
Therefore, 1280 litres of the 2% solution will have to be added to 640 litres of the 8% solution to create a mixture that is more than 4% but less than 6% boric acid.
Question:
Solve the inequalities and represent the solution graphically on number line. 5x+1>−24,5x−1<24
Answer:
Step 1: Subtract 1 from both sides of the inequality 5x + 1 > 24 5x > 25
Step 2: Divide both sides of the inequality by 5 x > 5
Step 3: Represent the solution graphically on a number line The solution is x > 5 and is represented by a dashed line on the number line starting at 5 and extending to the right.
Step 4: Solve the second inequality 5x  1 < 24 5x < 25
Step 5: Divide both sides of the inequality by 5 x < 5
Step 6: Represent the solution graphically on a number line The solution is x < 5 and is represented by a dashed line on the number line starting at 5 and extending to the left.
Question:
Solve the inequalities: 2≤3x−4≤5
Answer:

Add 4 to both sides: 2 + 4 ≤ 3x ≤ 5 + 4

Divide both sides by 3: 2/3 ≤ x ≤ 9/3
Question:
Solve the inequalities −15<3(x−2)/5≤0
Answer:
 3(x2)/5 ≤ 0
 3x6 ≤ 0
 3x ≤ 6
 x ≤ 2
 15 < x ≤ 2
Question:
IQ of a person is given by the formula IQ=MA/CA×100
Answer:
Step 1: Understand the formula
The formula states that the IQ of a person is equal to the Mental Age (MA) divided by the Chronological Age (CA) multiplied by 100.
Step 2: Substitute the values
If a person’s Mental Age is 20 and their Chronological Age is 10, then the formula can be rewritten as:
IQ = 20/10 × 100
Step 3: Calculate the IQ
IQ = 20/10 × 100 = 200
Question:
where MA is mental age and CA is chronological age. If 80≤IQ≤140 for a group 12 years old children, find the range of their mental age.
Answer:

IQ is a measure of intelligence, so the range of 80 to 140 indicates that the group of 12 year old children are of average intelligence.

Mental age is a measure of how mature a person is compared to their peers of the same age.

Therefore, the range of mental age for the group of 12 year old children is 12 years, as they are of average intelligence and the same age as their peers.
Question:
Solve the inequalities 7<(3x+11)/2≤11
Answer:
Step 1: 7<(3x+11)/2≤11
Step 2: 72<3x+11≤112
Step 3: 14<3x+11≤22
Step 4: 1411<3x≤2211
Step 5: 3<3x≤11
Step 6: 1<x≤11/3
Question:
Solve the inequality and represent the solution graphically on number line. 2(x−1)<x+5,3(x+2)>2−x
Answer:
Step 1: 2(x−1)<x+5
Step 2: 2x2<x+5
Step 3: x<7
Step 4: 3(x+2)>2−x
Step 5: 3x+6>2x
Step 6: 4x>8
Step 7: x>2
The solution is x > 2. The graph on the number line is shown below.
2––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> x
Question:
Solve the inequality and represent the solution graphically on number line. 5(2x−7)−3(2x+3)≤0,2x+19≤6x+47
Answer:

5(2x−7)−3(2x+3)≤0

10x−35−6x−9≤0

4x−44≤0

4x≥44

x≥11
The solution graphically on number line is x ≥ 11
Question:
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?
Answer:
Step 1: Calculate the amount of acid in 1125 litres of 45% solution.
45% of 1125 litres = 506.25 litres of acid
Step 2: Calculate the amount of acid required to make the resulting mixture contain more than 25% but less than 30% acid content.
30% of 1125 litres = 337.5 litres of acid
25% of 1125 litres = 281.25 litres of acid
Therefore, the amount of acid required is 337.5  281.25 = 56.25 litres of acid
Step 3: Calculate the amount of water to be added.
The amount of water to be added = 56.25 litres of acid  506.25 litres of acid = 450 litres of water
Therefore, 450 litres of water must 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.