Basic Terms and Terminology Relating to Translating Phrases and Sentences into Expressions, Equations and Inequalities

  • Equation: As the term suggests, an equation is a mathematical or arithmetic phrase that indicates the equality of two expressions separated with an equal (=) sign.
  • Expression: A series of mathematical or arithmetic symbols, numbers and/or variables that are grouped in a meaningful manner to have a value.
  • Equalities: Things that are equal
  • Inequalities: Not equal

Mathematical Symbols

Mathematical problems, equations, and expressions often entail the use of language and symbols. For this reason, you should be familiar with these terms and symbols, particularly those that are the most commonly used and encountered.

Symbol Name Read as Meaning Example
= equality equals, is equal to If x=y, x and y represent the same value or thing. 2+3=5
definition is defined as If x≡y, x is defined as another name of y (a+b)2≡a2+2ab+b2
approximately equal is approximately equal to If x≈y, x and y are almost equal. √2≈1.41
inequation does not equal, is not equal to If x≠y, x and y do not represent the same value or thing. 1+1≠3
< strict inequality is less than If x<y, x is less than y. 4<5
> strict inequality is greater than If x>y, x is greater than y. 3>2
strict inequality is much less than If x≪y, x is much less than y. 1≪999999999
strict inequality is much greater than If x≫y, x is much greater than y. 88979808≫0.001
inequality is less than or equal to If x≤y, x is less than or equal to y. 5≤6 and 5≤5
inequality is greater than or equal to If x≥y, x is greater than or equal to y. 2≥1 and 2≥2
proportionality is proportional to If x∝y, then y=kx for some constant k. If y=4x then y∝x and x∝y
+ addition plus x+y is the sum of x and y. 2+3=5
- subtraction minus x-y is the subtraction of y from x 5-3=2
x multiplication times x X y is the multiplication of x by y 4-5=20
/ division divided by x/y is the division of x by y 20/4=5
± plus-minus plus or minus x±y means both x+y and x-y The equation 3±√9 has two solutions, 0 and 6.
square root square root √x is a number whose square is x. √4=2 or -2
` summation sum over … from … to … of, sigma is the same as x1+x2+x3+xk
multiplication product over … from … to … of is the same as x1 * x2 * x3 * xk
! factorial factorial n! is the product 1*2*3...*n 5!=1*2*3*4*5=120
' material implication implies AB means that if A is true, B must also be true, but if A is false, B is unknown. x=3'x2=9, but x2=9'x=3 is false, because x could also be -3.
" material equivalence if and only if If A is true, B is true and if A is false, B is false. x=y+1"x-1=y
|…| absolute value absolute value of |x| is the distance along the real line (or across the complex plane) between x and zero |5|=5 and |-5|=5
|| parallel is parallel to If A||B then A and B are parallel
perpendicular is perpendicular to If A⊥B then A is perpendicular to B
... congruence is congruent to If A...B then shape A is congruent to shape B (has the same measurements)
φ golden ratio golden ratio The golden ratio is an irrational number equal to (1+√5)/2 or approximately 1.6180339887.
infinity infinity ∞ is a number greater than every real number.
{,} Set brackets the set of {a,b,c} is the set consisting of a, b, and c N={0,1,2,3,4,5}
N Natural numbers N N denotes the set of natural numbers {0,1,2,3,4,5...}

Expressions, Equations and Inequalities

An expression is a series of mathematical or arithmetic symbols, numbers and/or variables that are grouped in a meaningful manner to have a value.

Examples of Expressions

  • x
  • y
  • x -16
  • y + 22

An equation is a mathematical or arithmetic phrase that indicates the equality of two expressions separated with an equal (=) sign. The two expressions on either side of the equal sign are equal and equivalent.

Examples of Equations

  • x = 22
  • y = 689
  • x - 22 = 5 y
  • 8 (4x + 2) = 2 x - 7

An inequality is a mathematical or arithmetic phrase that is not equal; this inequality prohibits the equal sign. Instead, inequalities are expressed with greater than (>), less than (<), equal to or greater than (≥) or equal to or less than (≤).

Examples of inequalities

  • 54 - x > 44
  • 789 y < 900
  • 5 y + 67 ≥ 134
  • 6 x - 22 ≤ 100

On your TEAS examination and in everyday life, you will be expected to compose expressions, equations and inequalities from word problems. Below are some examples:

Composing Expressions

2 x + 4 is an expression. As you can see, you cannot solve expressions because their values cannot be determined because it is out of context with an equivalent and there is no = sign.

Composing Equations

You are practicing for your TEAS examination. At the current time you are able to answer 22 science questions in 20 minutes. How many science questions should you be able to answer in 45 minutes at this same rate?

Solution to this Word Problem:

This word problem indicates equality and an equation. It is not asking about an inequality such as greater than (>), less than (<), equal to or greater than (≥) or equal to or less than (≤).

The equation is set up and solved as below:

22 science questions : 20 minutes = x science questions : 45 minutes

20x = 33 x 45

20x = 990

x = 49.5

Composing Inequalities

Inequalities are set up in a similar manner to equations, however, an equal sign is not used. Instead, greater than (>), less than (<), equal to or greater than (≥) or equal to or less than (≤) are used.

Here are some word problems using an inequality:

Example 1

You have $16 saved in your wallet. How much more will you need to have more than $97.50?

$16 + x > $97.50

x > $97.50 - $16

x > $81.50

Answer: In this example, you will have to save more than $81.50 in order to have more than $97.50.

Example 2

You are trying to keep your grocery budget less than $125.00 per week. As you are shopping for groceries, you are adding up all of the individual items you have put in your basket. At the moment, you have $86.50 worth of groceries in your basket. How much more can you spend at the grocery store to keep your grocery budget less than $125.00 per week?

$86.50 + x < $125.00

x < $125.00 - $86.50

x < $38.50

Answer: You have to put less than $38.50 worth of groceries in the basket to keep your grocery budget less than $125.00 per week.

Example 3

You are practicing for your TEAS examination. At the current time you are able to answer 22 mathematics questions in 20 minutes. How many mathematics questions should you be able to answer in 45 minutes to exceed or meet your current rate?

20 x ≥ 22 x 45

20 x ≥ 990

x ≥ 49.5

Answer: In order to exceed or meet your current rate, you would have to answer at least 49.5, or 50 questions. You would meet your current rate if you can answer 50 questions in 45 minutes and you would exceed your current rate if you answer more than 50 questions in 45 minutes.

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