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## JSS1: MATHEMATICS - 2ND TERM

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Lesson 4, Topic 1
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# True or False Sentences

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A sentence can be true or false in a simple equation.

Such as $$\scriptsize \boxed {?} + 6 = 10$$

or   $$\scriptsize x \: + \: 6 = 10$$

This may be true or false depending on the value of $$\scriptsize \boxed {?} \: or \: x$$

If the value of the unknown is 4, the sentence above is True

if the the value of the unknown is 6, the sentence is false.

Also, if $$\scriptsize x \; or \; \boxed{?} = 4$$

then $$\scriptsize \boxed{?} + 10 = 15$$ is false since $$\scriptsize x \; or \; \boxed{?} = 4$$

$$\scriptsize 4 \; + \; 10 \neq 15$$

$$\scriptsize \neq$$ means “not equal to”

Note: A sentence which may be true or false is known as open sentence.

Example 1

Which of the following is true or false with the conditions given?

i. $$\scriptsize 3 \: \times \: \boxed {?} = 15$$ (when 5 replaces the box)

ii. $$\scriptsize 2a \: + \: 10 = 20$$ (when a  =  12)

iii. $$\scriptsize a \: + \: 6 = 8$$ (when  a   =  2)

iv. $$\scriptsize 9x \: – \: 10 = 6$$ (when  x  =   3)

Solution

i. $$\scriptsize 3 \: \times \: \boxed {?} = 15$$ (when 5 replaces the box)

When $$\scriptsize \boxed {?} = 5$$

$$\scriptsize 3 \: \times \: \boxed {5} = 15$$

$$\scriptsize 3 \: \times \: 5 = 15$$ is true because both sides are equal.

ii. $$\scriptsize 2a \: + \: 10 = 20$$ (when a  =  12)

$$\scriptsize \therefore 2 \: \times \: 12 \: + \: 10 = 20$$

$$\scriptsize 24 \: + \: 10 = 20$$

$$\scriptsize 34 \neq 20$$

$$\scriptsize \therefore 2a \: + \: 10 = 20$$ is false when a  =  12

iii. $$\scriptsize a \: + \: 6 = 8$$ (when  a   =  2)

2   +  6   =  8   The sentence is true when a  =  2

iv. $$\scriptsize 9x \: – \: 10 = 6$$ (when  x  =   3)

Let’s substitute x = 3 into $$\scriptsize 9x \: – \: 10$$

$$\scriptsize 9x \: – \: 10$$

$$\scriptsize = (9 \: \times \: 3) \; – \: 10$$

$$\scriptsize = 27 \: – \: 10 \\ \scriptsize = 17$$

$$\scriptsize 9x \: – \: 10 \neq 6 \: \\ \scriptsize ….\: because \: 17 \neq 6$$

Hence  9x   –  10   =  6  is false when x  =  3
when x = 3, 9x   –  10   =  17

Example 2

Find the values of the unknown that will make the following sentences true.

i. $$\scriptsize a \: + \: 5 = 9$$

ii. $$\scriptsize 3x = 30$$

iii. $$\frac{y}{2} \scriptsize = 7$$

iv. $$\scriptsize w \; – \: 8 = 6$$

Solution

i. $$\scriptsize a \: + \: 5 = 9$$

“a”  means a number added to 5 to make 9.

By observation, 4 is the only number.

i.e

$$\scriptsize a \: + \: 5 = 9$$

$$\scriptsize 4 \: + \: 5 = 9$$

a   =  4 is  true

ii. $$\scriptsize 3x = 30$$

x means the number you can multiply by 3 to obtain 30.

By inspection, 10 is the number because 3 x 10 = 30

By calculation;

$$\scriptsize 3x = 30$$

Divide both sides by 3

⇒ $$\frac{3x}{3} = \frac{30}{3} \\ = \frac{\not {3}x}{\not {3}} = \frac{30}{3} \\ \scriptsize x = \normalsize \frac{30}{3}\\ \scriptsize x = 10$$

x = 10 is true

iii. $$\frac{y}{2} \scriptsize = 7$$

y means a number that 2 can divide to obtain 7 with no remainder.

By inspection, the number is 14

$$\scriptsize y \: \div \: 2 = 7$$

$$\scriptsize 14 \: \div \: 2 = 7$$

y = 14 is true

iv. $$\scriptsize w \; – \: 8 = 6$$

w means a number that when 8 is subtracted from we get 6. By inspection. 14 is the number.

$$\scriptsize w \; – \: 8 = 6$$

$$\scriptsize 14 \; – \: 8 = 6$$

w = 14 is true