~ Lovers of Math

Equations and Inequalities

In algebra, an equation is a mathematical expression that contains an equals sign. It tells us that two expressions represent the same number. For example, $y = 12x$ is an equation. An inequality is a mathematical expression that contains inequality signs. For example, $y \le 12x$ is an inequality. Inequalities are used to tell us that an expression is either larger or smaller than another expression. Equations and inequalities can contain both variables and constants.

Variables are usually given a letter and they are used to represent unknown values. These quantities can change because they depend on other numbers in the problem.

Constants are quantities that remain unchanged. Ordinary numbers like $2, \ -3, \ \frac{3}{4},$ and $\pi$ are constants.

Equations and inequalities are used as a shorthand notation for situations that involve numerical data. They are very useful because most problems require several steps to arrive at a solution, and it becomes tedious to repeatedly write out the situation in words.

Write Equations and Inequalities

Here are some examples of equations:
$3x - 2 = 5 \qquad x + 9 = 2x + 5 \qquad \frac{x}{3} = 15 \qquad x^2 + 1 = 10$
To write an inequality, we use the following symbols:
> greater than
$\ge$ greater than or equal to
< less than
$\le$ less than or equal to
$\neq$ not equal to
Here are some examples of inequalities:

$3x <5 \qquad 4 - x \le 2x \qquad x^2 + 2x - 1 > 0 \qquad \frac{3x}{4} \ge \frac{x}{2} - 3$

The most important skill in algebra is the ability to translate a word problem into the correct equation or inequality so you can find the solution easily. The first two steps are defining the variables and translating the word problem into a mathematical equation.

Defining the variables means that we assign letters to any unknown quantities in the problem.

Translating means that we change the word expression into a mathematical expression containing variables and mathematical operations with an equal sign or an inequality sign.

Example

Define the variables and translate the following expressions into equations.
a) A number plus 12 is 20.
b) 9 less than twice a number is 33.
c) $20 was one quarter of the money spent on the pizza.

Solution

a) Define
Let $n=$ the number we are seeking.
Translate
A number plus 12 is 20.
$n + 12 = 20$
b) Define
Let $n=$ the number we are seeking.
Translate
9 less than twice a number is 33.
This means that twice the number, minus 9, is 33.
$2n - 9 = 33$
c) Define
Let $m =$ the money spent on the pizza.
Translate
$20 was one quarter of the money spent on the pizza.
$20 = \frac{1}{4} m$
Often word problems need to be reworded before you can write an equation.

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