Algebra

Number Lines

Big Picture

Real numbers can be visualized using a number line. Number lines are helpful for picturing a number of algebra concepts, including opposite numbers, operations of real numbers, and inequalities.

Key Terms

Number Line: A straight line where every point on the line represents a real number

Opposite Numbers: Two numbers that are the same distance from zero but on opposite sides of the number line.

Absolute Value: The absolute value of a number is the distance of that number from 0.

Graphing Rational Numbers

All real numbers can be drawn on a number line. A number line plots the greatest number to the farthest right, and the least to the farthest left.

  • The numbers increase from left to right
  • 2 is further to the right than -5 is, so 2 is greater than -5. 2 > -5
Rational numbers

The number line above is divided into intervals of 1, so the numbers on the line increase by +1 from left to right. A number line can be divided into as many sub-intervals as you need. To plot \(\frac{2}{3}\) on the number line, you can have sub-intervals of \(\frac{2}{3}\)

Rational numbers

A number line represents all real numbers (that’s an infinite amount of numbers)!

  • The arrows on both ends of the number line will keep extending in length forever. This means 100 and -25 are a part of the number line above.
  • Numbers that are not labeled on the number line are also included. Numbers like 1.111, -4/3, and \(\sqrt{2}\) are a part of the number line.

Opposite Numbers

Opposite numbers are on opposite sides of 0 and are the same distance away from zero.

Opposite numbers
  • Every number has an opposite - the opposite number of 0 is 0
  • The sum of a number and its opposite is always zero
  • Example: \(2 + -2 = 0, -5.7 + 5.7 = 0\)
  • The opposite of a number is sometimes called the additive inverse
  • Multiplying a number by -1 is the same as finding the opposite number
  • The opposite number of an expression can be found by multiplying the entire expression by -1
  • Example: opposite of \((x - 2) ≠ (x + 2)\); instead the opposite of \((x - 2) = -(x - 2) = (-x + 2) = (2 - x)\)

Algebra

Number Lines cont.

Absolute Value

The absolute value of a number is its distance from zero.

  • Every whole number is a rational number where the denominator equals 1.
  • The absolute values of opposite numbers are equal
  • Example: \(|-4| = |4| = 4\) because both -4 and 4 are 4 units away from 0

Operations

Add Using a Number Line

To add numbers on a number line, start on the first number in the expression. If you are adding a positive number, then move to the right by the number of units equal to the next number in the expression.

Example:\( -2 + 3\)

  • Start at -2
  • End at 1, so -2 + 3 = 1

Subtracting Using a Number Line

To subtract numbers on a number line, move left instead of right. Subtracting a number is like adding a negative number.

Example: \(2 - 3 = 2 + (-3)\)

  • Start at -2
  • Move to the left by 3 units
  • End at 1, so \(-2 + 3 = 1\)

If a number is negative, move left, and if a number is positive, move right.

Inequalities

Inequalities in one variable can be shown on a number line.

Symbols for Inequalities

Symbol
Meaning
<
less than
less than or equal to
>
greater than
greater than or equal to

If we want to show all the values of x that is greater than some number a, we are looking for all the numbers of x that will make x > a true. The number line would look like:

Opposite numbers
  • The open circle at a means that a is not included
  • The shaded arrow includes all the numbers to the right of a

If we want to show x ≥ a, then we would want to include a. If x=a, then the inequality would still be true.

Opposite numbers
  • The filled circle at a means that a is included

Showing less than (<) or less than and equal (≤) is similar, except the arrow would now point to the left.

Opposite numbers
Opposite numbers