True or False Sentences

What is a Mathematical Equation:

A mathematical equation is an expression containing at least one variable (an unknown value and an “equal sign”  = )   with a mathematical expression on each side of it.

An equation can be as simple as; a =  0

e.g.   

x   +  2  =  0
2x  –  8  =  0   
x  –  1  =  3
x  +  5  =  15

True or False Sentences:

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 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 an open sentence.

Example 4.1.1:

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 4.1.2:

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