math
Example: -6 < 7
-6 - 5 < 7 - 5 -11 < 2
Example: 10 > 4
10 + 20 > 4 + 20 30 > 24
2+2+2+2
2 to the 4
Example: 6 < 10
6 + 7 < 10 + 7 13 < 17
numerical sentences
Expressions such as 2 • 6 = 12 and 6 > 4 6 > 10 is a numerical sentence; however, it is a false statement. Another example of a false statement is 8 = 5.
Solve -5 + n = 9 for n.
Solution: Since adding a number and its opposite results in zero, add 5 to both sides of the equation. -5 + n = 9 -5 + 5 + n = 9 + 5 0 + n = 14 n = 14 Check: -5 + 14 9 9 = 9 (True) The solution set is {14}.
Solve p + (-7) = -12 for p.
Solution: p + (-7) = -12 is the same as p - 7 = -12 p - 7 + 7 = -12 + 7 p = -5 Check: -5 + (-7) -12 -12 = -12 (True) The solution set is {-5}.
set
a collection or group of objects indicated by braces, { }
equation
a mathematical sentence whose verb is 'equal' (=)
element
a member of a set
null set
a set containing no elements; also called the empty set
solution
a value or values of the variable that make an algebraic sentence true
open sentence
an equation that contains a variable or variables
Solving an inequality
means finding the values of the variable that make the inequality true. We solve inequalities in much the same way as we solved equations. Generally, an equation has one element in the solution set, whereas solution sets of inequalities are infinite.
subset
set A is a subset of set B if all of the elements of set A are contained in set B or it is the empty set
additive identity
the number 0 (for the real numbers)
multiplicative identity
the number 1 (for the real numbers)
multiplicative inverse
the reciprocal of a number; if x is a number other than zero, then 1/x is the multiplicative inverse; the product of a number and its multiplicative inverse is 1
symmetric property
when the sides of an equation are interchanged, an equivalent equation is formed; if a = b, then b = a
The addition property of inequality
you may add (or subtract) any number you choose to both sides of an inequality without changing the order of the inequality.