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# Lesson 6.2: The Horizontal Bar - Repeating Decimals, Greater than or equal to, and Less than or equal to

## Symbols

$\text{directly over symbol}$

$\text{horizontal bar}$

$\text{≥ greater than or equal to}$
⠨⠂⠱

$\text{≤ less than or equal to}$
⠐⠅⠱

## Explanation

### Horizontal Bar and Repeating Decimals

As discussed before, shapes may need to be modified for special purposes. Such is the case when a shape needs to be placed above or below another symbol. The use of a bar is one example where this occurs. The bar, dots one five six, is often used to show the repeating portion of decimals to indicate a sample mean in statistics, or as a line segment in geometry. When used with decimals, the raised bar indicates that the digit or series of digits covered by the bar repeat an infinite number of times over and over.

If the expression contains a single digit that repeats an infinite number of times, the digit is followed by the symbol for a horizontal bar, dots one five six. If, however, there is more than one digit in the expression or if the bar is above more than one digit, then you must use the rules for modifying an expression. First, the multipurpose indicator, dot five, must be used directly before the first digit under the bar. Next, you should input the expression or digits that are covered by the bar. Use either the directly over (dots one two six) or the directly under (dots one four six) indicator. For the purposes of these problems, since the bar indicating the repeating decimal is above the numbers, you would use the directly over indicator. The next symbol is the symbol you wish to place over the expression. In this case it is the horizontal bar, dots one five six. Finally use the termination indicator (dots one two four five six) after the last number that is covered by the horizontal bar.

### Example 1

$\stackrel{‾}{x}$
⠭⠱

### Example 2

$\stackrel{‾}{x}=53$
⠭⠱⠀⠨⠅⠀⠼⠢⠒

### Example 3

$.\stackrel{‾}{5}$
⠼⠨⠢⠱

### Example 4

$.\stackrel{‾}{18}$
⠼⠨⠐⠂⠦⠣⠱⠻

### Example 5

$5.8\stackrel{‾}{144}$
⠼⠢⠨⠦⠐⠂⠲⠲⠣⠱⠻

### Less Than or Equal to and Greater Than or Equal to

You may also see a horizontal bar used with a sign of comparison such as greater than or less than. In print, the vertical bar is usually shown below the sign of comparison. When a vertical bar is shown with a less than or greater than sign, they are then read as greater than or equal to and less than or equal to. In braille, the horizontal bar is shown after these signs of comparison.

Remember greater than and less than are two-cell symbols. Greater than is made of dots four six, followed by a dot two. Less than is made of dot five followed by dots one and three. To get greater than or equal or less than or equal to, simply put the braille symbol for a horizontal bar after the greater than or less than symbol. These will now be three-cell symbols. You will still need to leave a blank space before and after a greater than or equal to and a less than or equal to since they are still signs of comparison.

### Example 6

$5\le 7$
⠼⠢⠀⠐⠅⠱⠀⠼⠶

### Example 7

$7\le 7$
⠼⠶⠀⠐⠅⠱⠀⠼⠶

### Example 8

$9\ge 5$
⠼⠔⠀⠨⠂⠱⠀⠼⠢

### Example 9

$9\ge 9$
⠼⠔⠀⠨⠂⠱⠀⠼⠔

### Example 10

$x+5\le 10$
⠭⠬⠢⠀⠐⠅⠱⠀⠼⠂⠴

### Example 11

$|-6|\ge 3$
⠳⠤⠖⠳⠀⠨⠂⠱⠀⠼⠒

### Example 12

$\frac{3}{4}\le △$
The triangle represents an omitted value.
⠹⠒⠌⠲⠼⠀⠐⠅⠱⠀⠀⠫⠞