• Use interval notation to show solutions to inequalities.
• Solve linear inequalities by applying their properties.

Using Interval Notation

In many academic institutions, student performance is evaluated using a Grade Point Average (GPA) system where scores range from [latex]0[/latex] to [latex]4[/latex]. This scale can be represented using linear inequalities to define the complete range of possible GPAs. How could these GPA scores be expressed mathematically?

• Set-Builder Notation: [latex]\{x \mid 0 \leq x \leq 4\}[/latex], where [latex]x[/latex] represents a student’s GPA.
  The inequality [latex]0 \leq x \leq 4[/latex] encapsulates the idea that a GPA can be as low as [latex]0[/latex], which might indicate failing grades or no coursework completed, and as high as [latex]4[/latex], representing perfect academic performance.
• Interval Notation: [latex][0, 4][/latex].
  This notation correctly specifies that the GPA can be any value including and between [latex]0[/latex] and [latex]4[/latex].
• Number Line: The number line below illustrates that the solution set includes all real numbers between [latex]0[/latex] and [latex]4[/latex].
  

  

Interval Notation:

• Parentheses represent solutions greater or less than the number.
• Brackets represent solutions that are greater than or equal to or less than or equal to the number.
• Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.”

What is an Inequality?An inequality is a mathematical statement that compares two expressions using symbols such as:

• [latex]<[/latex] (less than)
• [latex]>[/latex] (greater than)
• [latex]\leq[/latex] (less than or equal to)
• [latex]\geq[/latex] (greater than or equal to)

Inequalities show that one side of the statement is not equal to the other. Instead, it describes a range of possible values. For example, the inequality [latex]x > 3[/latex] means that [latex]x[/latex] can be any number greater than 3.

Inequality	Words	Interval Notation	Number Line
[latex]{a}\lt{x}\lt{ b}[/latex]	all real numbers between a and b, not including a and b	[latex]\left(a,b\right)[/latex]
[latex]{x}\gt{a}[/latex]	All real numbers greater than a, but not including a	[latex]\left(a,\infty \right)[/latex]
[latex]{x}\lt{b}[/latex]	All real numbers less than b, but not including b	[latex]\left(-\infty ,b\right)[/latex]
[latex]{x}\ge{a}[/latex]	All real numbers greater than a, including a	[latex]\left[a,\infty \right)[/latex]
[latex]{x}\le{b}[/latex]	All real numbers less than b, including b	[latex]\left(-\infty ,b\right][/latex]
[latex]{a}\le{x}\lt{ b}[/latex]	All real numbers between a and b, including a	[latex]\left[a,b\right)[/latex]
[latex]{a}\lt{x}\le{ b}[/latex]	All real numbers between a and b, including b	[latex]\left(a,b\right][/latex]
[latex]{a}\le{x}\le{ b}[/latex]	All real numbers between a and b, including a and b	[latex]\left[a,b\right][/latex]
[latex]{x}\lt{a}\text{ or }{x}\gt{ b}[/latex]	All real numbers less than a or greater than b	[latex]\left(-\infty ,a\right)\cup \left(b,\infty \right)[/latex]
All real numbers	All real numbers	[latex]\left(-\infty ,\infty \right)[/latex]

Using interval notation to express the inequalities.

1. All real numbers greater than or equal to [latex]-2[/latex].
2. All real numbers between and including [latex]-3[/latex] and [latex]5[/latex].
3. All real numbers less than or equal to [latex]-1[/latex] or greater than or equal to [latex]1[/latex].
4. All real numbers less than [latex]-2[/latex] or greater than or equal to [latex]3[/latex].

Show Answer

1. [latex]\left[-2,\infty \right)[/latex]
2. [latex]\left[-3,5\right][/latex]
3. [latex]\left(-\infty ,-1\right]\cup \left[1,\infty \right)[/latex]
4. [latex]\left(-\infty ,-2\right)\cup \left[3,\infty \right)[/latex]

Represent the number line below in words, inequality, and an interval notation.

Show Answer

• Words: all real numbers less than [latex]10[/latex]
• Interval Notation: [latex](-\infty, 10)[/latex]
• Inequality: [latex]x<10[/latex]