Indicating the solution to an inequality much as [latex]x\ge 4[/latex] buttocks be achieved in several ways.
We can use a number line equally shown in Figure 2. The blue ray begins at [latex]x=4[/rubber-base paint] and, as indicated by the arrowhead, continues to eternity, which illustrates that the solution set includes all real Numbers greater than or equal to 4.
Figure 2
We can use readiness-builder notation: [rubber-base paint]\{x|x\ge 4\}[/rubber-base paint], which translates to "all real numbers racket x so much that x is greater than or equal to 4." Notice that braces are used to indicate a set.
The third method acting is interval notation, in which root sets are indicated with parentheses or brackets. The solutions to [latex]x\ge 4[/latex] are represented as [rubber-base paint]\left[4,\infty \right)[/latex paint]. This is perhaps the most reclaimable method, arsenic IT applies to concepts unnatural later in this course and to other higher-level math courses.
The main concept to retrieve is that parentheses represent solutions greater OR to a lesser degree the number, and brackets represent solutions that are greater than or adequate or less than or quits to the number. Use parentheses to represent infinity or electronegative infinity, since positive and negative eternity are not numbers in the habitual sense of the word and, therefore, cannot be "equaled." A a couple of examples of an interval, operating theater a set of numbers in which a solution falls, are [latex]\left[-2,6\right)[/latex], or all numbers pool between [latex]-2[/latex] and [latex]6[/latex], including [latex]-2[/rubber-base paint], but not including [latex]6[/latex]; [latex]\left(-1,0\right)[/latex], all real numbers 'tween, but non including [latex]-1[/latex] and [latex]0[/latex]; and [latex]\left(-\infty ,1\right][/latex], all real numbers pool to a lesser degree and including [latex]1[/latex]. The table below outlines the possibilities.
| Set Indicated | Put-Builder Notation | Interval Note |
|---|---|---|
| All real numbers betwixt a and b, but non including a OR b | [latex]\{x|a<x<b\}[/latex] | [latex paint]\left(a,b\right)[/latex] |
| All real numbers game greater than a, but not including a | [latex]\{x|x>a\}[/latex] | [latex]\left hand(a,\infty \right)[/latex] |
| All real numbers racket fewer than b, but not including b | [latex]\{x|x<b\}[/latex] | [latex]\left(-\infty ,b\right)[/latex paint] |
| All real numbers racket greater than a, including a | [latex]\{x|x\ge a\}[/latex] | [latex]\left[a,\infty \right)[/latex] |
| All real numbers less than b, including b | [latex]\{x|x\le b\}[/latex] | [latex]\socialist(-\infty ,b\right][/rubber-base paint] |
| All real numbers between a and b, including a | [rubber-base paint]\{x|a\le x<b\}[/latex] | [latex]\left[a,b\right)[/latex] |
| All real numbers between a and b, including b | [latex]\{x|a<x\le b\}[/latex] | [latex]\left(a,b\right][/latex] |
| All real numbers between a and b, including a and b | [latex]\{x|a\le x\lupus erythematosus b\}[/rubber-base paint] | [rubber-base paint]\left[a,b\right][/latex] |
| All tangible numbers to a lesser degree a or greater than b | [latex]\{x|x<a\text{ and }x>b\}[/latex] | [latex]\leftover(-\infty ,a\right)\cup \left(b,\infty \right)[/latex] |
| All real numbers | [latex]\{x|x\text{ is all real Numbers}\}[/latex] | [latex]\left(-\infty ,\infty \right)[/latex] |
Model 1: Using Interval Notation to Express All Real Numbers pool Greater Than operating theatre Equal to a
Use time interval notation to indicate all real numbers racket greater than or equal to [latex]-2[/latex].
Solution
Utilise a bracket on the left of [latex]-2[/latex] and parentheses after eternity: [latex]\left[-2,\infty \right)[/latex]. The bracket indicates that [latex]-2[/latex] is included in the set with all real numbers greater than [rubber-base paint]-2[/latex paint] to infinity.
Try IT 1
Use time interval notation to signal all real numbers between and including [latex]-3[/latex] and [latex]5[/latex].
Solution
Lesson 2: Using Interval Annotation to Express Each Real Numbers Little Than Oregon Even to a or Greater Than or Isothermal to b
Write the interval expressing all genuine numbers less than or same to [rubber-base paint]-1[/latex] or greater than or equal to [latex]1[/latex].
Solution
We cause to write two intervals for this example. The first interval must argue all realistic numbers to a lesser degree or equal to 1. So, this interval begins at [latex]-\infty [/latex] and ends at [latex paint]-1[/latex], which is scrivened atomic number 3 [rubber-base paint]\left(-\infty ,-1\right][/latex].
The second interval must show all real numbers greater than operating theatre peer to [latex]1[/rubber-base paint], which is written every bit [latex]\left[1,\infty \right)[/rubber-base paint]. However, we want to combine these 2 sets. We accomplish this by inserting the union symbol, [latex]\loving cup [/latex], between the ii intervals.
[rubber-base paint]\left(-\infty ,-1\flop]\cup \nigh[1,\infty \right)[/latex]
Try It 2
Express completely real numbers less than [latex]-2[/rubber-base paint] or greater than operating room equal to 3 in interval notation.
Solution
x greater than or equal to 2 interval notation
Source: https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/using-interval-notation/
