Mathematics Tutorials: Maths Made Easy!: 2013

Linear Inequalities

An inequality is just like linear equation which has two sides and an unknown variable which are always represented with alphabets (a-z) but deals with a lot of signs.

Signs of inequalities

> Greater than

< Less than

≥ Greater or equal to

≤ Less than or equal to

Solve the following inequalities;

Example 1; Simplify 2x – 4 > 8

Solution

Note;

In order to do this, you must consider the two sides of this inequality.

2x – 4 > 8 (Make x the subject of the equation)

2x > 8 + 4

2x > 12 (Divide both sides by 2)

x > 6.

Example 2; Solve the inequalities 3x – 8 ≤ 10 + 5x

Solution

3x – 8 ≤ 10 + 5x (Collect like terms)

3x – 5x ≤ 10 + 8

- 2x ≤ 18 (Divide both sides by the co-efficient of x which is -2)

x ≥ - 9.

Note; the inequality sign must change when dividing with a minus sign).

Practice these questions below;

1. Simplify 4x – ¹/₃ ≥ ²/_3x + 2

3. Simplify 6x – 2 < 2x + 8

4. Simplify ¹/_3y > ¹/_2y + ¹/₄

5. Simplify ⁶/_x ≥ 2

Word Problems Leading To Linear Inequalities

Words problems leading to inequalities are handled in a similar fashion like the linear equations except that the equality sign (=) changes to inequality sign.

Examples;

1. One-fifth of a number added to two-third of the same number is greater than 26. Find the range of values of the number.

Solution

Step 1

Let x represent the unknown number

The mathematical interpretation of the question will be;

^x/₅ + ^2x/₃ > 26

Step 2

Multiply through by 15 which is the L.C.M of 5 and 3;

3x + 10x > 390

13x > 390

Step 3

Divide both sides by 13;

x > ³⁹⁰/₁₃

x > 30.

Therefore, the range of values of x will be numbers greater than 30, which are 31, 32, 33, 34, 35 …

2. Ade bought x biros at #8 each and (x + 4) rulers at #20 each. He spent less than #200. Form an inequality for the statement and solve it.

Solution

Step 1

Interpret the statement;

The biros cost #8x …….(1)

The rulers cost #20(x +4) …….(2)

Step 2

Expand the second statement and you will have;

(#20x + #80)

Step 3

Combining (1) and (2) gives the total cost of both biros and rulers;

#(8x + 20x + 80) which is less than (<) #200.

Step 4

Forming an inequality equation below, you’ll have;

8x + 20x + 80 < 200

Step 5

Re-arrange and collect like terms;

Hence 28x < 200 – 80 (This is the inequality statement)

28x < 120

Step 6

Divide both sides by 28;

x < ¹²⁰/₂₈

Therefore, x < 4.29.

Practice these questions below;

1. The sum of twice a number and 5 is less than the sum of one-third of the number and 6.

2. Two-thirds of a certain number is greater than the sum of the number and 6.

3. The sum of twice a number and 15 is less than thrice the same number minus 9.

4. A cyclist travels xkm in 4 hours, then (x + 60)km in 7 hours. Its average speed does not exceed 150km/h.

5. A boy bought x mangoes at #5 and 3x oranges at #6. He collected some balance from #30.

Ratio and Percentage

A ratio is a numerical method of comparing two quantities of the same type, such as sum of money, length, weight, ages, marks, etc…

As ratio is a comparison of relative magnitude of two quantities, it may not necessarily contain units. It may be in fraction or separated by column in between them. For example; ^P/_Q or P:Q which is pronounced as P ratio Q. A ratio is always numbers.

The following rules should be implemented when dealing with ratios.

Rule 1; If ^a/_b and ^c/_d are two different ratios. Let ^a/_b = ^c/_d

Therefore, ad = bc (when cross multiplied)

Rule 2; If ^a/_b = ^c/_d

Therefore, ^b/_a = ^d/_c (alternating the ratios)

Rule 3; If ^a/_b = ^c/_d

Therefore, ^a/_c = ^b/_d (inverting the ratios)

Examples;

1. In a certain class, the ratio of boys to girls is 2:5. If there are 40 boys, find how many girls are there.

Solution

Let the no of girls be x

Then 2:5 = 40:x

²/₅ = ⁴⁰/_x (cross multiply)

2x = 40 X 5

2x = 200 (divide both sides by 2)

x = ²⁰⁰/₂

Therefore, x = 100.

This means that there are 100 girls in the class.

2. Which ratio is greater, 6:8 or 12:22?

Solution

6:8 is the same as ⁶/₈.

Divide 6 by 8 and the answer is 0.75.

While 12:22 is the same as ¹²/₂₂.

Divide 12 by 22 and the answer is 0.545.

Which is bigger between 0.75 and 0.545?

Obviously it is 0.75.

Therefore this means that 6:8 is greater than 12:22.

3. Decrease #110.81 in the ratio 4:7.

Solution

In order to decrease #110.81, multiply it by ⁴/₇.

#110.81 X ⁴/₇

This can be written as;

= #63.92 (final answer)

Practice these questions below;

1. The ratio of the circumference of a circle to its diameter is 22:7. What is the circumference of a circle of diameter 15.6m?

2. In each class, find which of the two ratios is greater;

(a) 16 : 7 or 17 : 6

(b) 2.5g : 2kg or 0.4kg : 300kg

3. Divide 490 in the ratio 2:5:7.

4. In preparing a recipe for cake, the ratio of the flour to sugar is 40:3. Find the required amount of flour to 18kg of sugar.

5. A worker’s income is increased in the ratio 36:30. Find the increase percent.