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)