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Study Guides > 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
\neq not equal to 28{2}\neq{8}, 2 is not equal to 8
>\gt greater than 5>1{5}\gt{1}, 5 is greater than 1
<\lt less than 2<11{2}\lt{11}, 2 is less than 11
\geq greater than or equal to 44{4}\geq{ 4}, 4 is greater than or equal to 4
\leq less than or equal to 79{7}\leq{9}, 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
a<x<b{a}\lt{x}\lt{ b} all real numbers between a and b, not including a and b (a,b)\left(a,b\right)
x>a{x}\gt{a} All real numbers greater than a, but not including a (a,)\left(a,\infty \right)
x<b{x}\lt{b} All real numbers less than b, but not including b (,b)\left(-\infty ,b\right)
xa{x}\ge{a} All real numbers greater than a, including a [a,)\left[a,\infty \right)
xb{x}\le{b} All real numbers less than b, including b (,b]\left(-\infty ,b\right]
ax<b{a}\le{x}\lt{ b} All real numbers between a and b, including a [a,b)\left[a,b\right)
a<xb{a}\lt{x}\le{ b} All real numbers between a and b, including b (a,b]\left(a,b\right]
axb{a}\le{x}\le{ b} All real numbers between a and b, including a and b [a,b]\left[a,b\right]
x<a or x>b{x}\lt{a}\text{ or }{x}\gt{ b} All real numbers less than a or greater than b (,a)(b,)\left(-\infty ,a\right)\cup \left(b,\infty \right)
All real numbers All real numbers (,)\left(-\infty ,\infty \right)
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
a>ba>b cc ac>bcac>bc
5>35>3 33 15>915>9
a>ba>b c-c ac<bc-ac<-bc
5>35>3 3-3 15<9-15<-9
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
a>ba>b cc ac>bc \displaystyle \frac{a}{c}>\frac{b}{c}
4>24>2 22 42>22 \displaystyle \frac{4}{2}>\frac{2}{2}
a>ba>b c-c ac<bc \displaystyle -\frac{a}{c}<-\frac{b}{c}
4>24>2 2-2 42<22 \displaystyle -\frac{4}{2}<-\frac{2}{2}

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