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Guides d'étude > ALGEBRA / TRIG I

Summary: Solving Single- and Multi-Step Inequalities

Key Concepts

Inequality Signs

The box below shows the symbol, meaning, and an example for each inequality sign.
Symbol Words Example
[latex]\neq [/latex] not equal to [latex]{2}\neq{8}[/latex], 2 is not equal to 8
[latex]\gt[/latex] greater than [latex]{5}\gt{1}[/latex], 5 is greater than 1
[latex]\lt[/latex] less than [latex]{2}\lt{11}[/latex], 2 is less than 11
[latex] \geq [/latex] greater than or equal to [latex]{4}\geq{ 4}[/latex], 4 is greater than or equal to 4
[latex]\leq [/latex] less than or equal to [latex]{7}\leq{9}[/latex], 7 is less than or equal to 9
The table below describes all the possible inequalities that can occur and how to write them using interval notation, where a and b are real numbers.
Inequality Words Interval Notation
[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]
The following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:
Start With Multiply By Final Inequality
[latex]a>b[/latex] [latex]c[/latex] [latex]ac>bc[/latex]
[latex]5>3[/latex] [latex]3[/latex] [latex]15>9[/latex]
[latex]a>b[/latex] [latex]-c[/latex] [latex]-ac<-bc[/latex]
[latex]5>3[/latex] [latex]-3[/latex] [latex]-15<-9[/latex]
The following table illustrates how the division property is applied to inequalities, and how dividing by a negative reverses the inequality:
Start With Divide By Final Inequality
[latex]a>b[/latex] [latex]c[/latex] [latex] \displaystyle \frac{a}{c}>\frac{b}{c}[/latex]
[latex]4>2[/latex] [latex]2[/latex] [latex] \displaystyle \frac{4}{2}>\frac{2}{2}[/latex]
[latex]a>b[/latex] [latex]-c[/latex] [latex] \displaystyle -\frac{a}{c}<-\frac{b}{c}[/latex]
[latex]4>2[/latex] [latex]-2[/latex] [latex] \displaystyle -\frac{4}{2}<-\frac{2}{2}[/latex]

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