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Solving Linear Inequalities Help

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By — McGraw-Hill Professional
Updated on Sep 27, 2011

Introduction to Solving Linear Inequalities

Linear inequalities are solved much the same way as linear equations with one exception: when multiplying or dividing both sides of an inequality by a negative number the inequality sign must be reversed For example 2 < 3 but –2 > –3. Adding and subtracting the same quantity to both sides of an inequality never changes the direction of the inequality sign.

Examples

Solving Linear Inequalities Examples

Solving Linear Inequalities

Solving Linear Inequalities

Find practice problems and solutions at Solving Linear Inequalities Practice Problems - Set 1.

Interval Notation

The symbol for infinity is “∞,” and “−∞” is the symbol for negative infinity. These symbols mean that the numbers in the interval are getting larger in the positive or negative direction. The intervals for the previous examples and practice problems are called infinite intervals.

An interval consists of, in order, an open parenthesis “(” or open bracket “[,” a number or “–∞,” a comma, a number or “∞,” and a closing parenthesis “)” or closing bracket “].” A parenthesis is used for strict inequalities ( x < a and x > a) and a bracket is used for an “or equal to” inequality ( xa and xa ). A parenthesis is always used next to an infinity symbol.

Inequality

Interval

x < number

(−∞, number)

x > number

(number, ∞)

x ≤ number

(−∞, number]

x ≥ number

[number, ∞)

Examples

Solving Linear Inequalities

The table below gives the relationship between an inequality, its region on the number line, and its interval notation.

Inequality

Number Line Region

Interval Notation

x < a

Solving Linear Inequalities

(−∞, a )

xa

Solving Linear Inequalities

(−∞, a ]

x > a

Solving Linear Inequalities

( a , ∞)

xa

Solving Linear Inequalities

[ a , ∞)

Ordinarily the variable is written on the left in an inequality but not always. For instance to say that x is less than 3 ( x < 3) is the same as saying 3 is greater than x (3 > x ).

Inequality

Equivalent

Inequality

x < a

a > x

xa

ax

x > a

a < x

xa

ax

 

Find practice problems and solutions at Solving Linear Inequalities Practice Problems - Set 2.

More practice problems for this concept can be found at: Algebra Linear Inequalities Practice Test.

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