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Math Lesson- Solving Linear Inequalities
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29 Sep 2016, 12:16
29 Sep 2016, 12:16
Expert Reply
Solving Linear Inequalities
Rules of Operations in an Inequality
1. Multiplication/Division Properties for Inequalities: when multiplying/dividing by a positive value
If a < b AND c is positive, then \(ac < bc\)
If a < b AND c is positive, then \(a/c < b/c\)
2. Multiplication/Division Properties for Inequalities: when multiplying/dividing by a negative value
If a < b AND c is negative, then ac > bc
If a < b AND c is negative, then a/c > b/c
When solving inequalities, if you multiply or divide through by a negative, you must also flip the inequality sign.
3. Compound Inequality
Solve 10 < 3x + 4 < 19.
This is what is called a "compound inequality". It works just like regular inequalities, except that it has three "sides". So, for instance, when I go to subtract the 4, I will have to subtract it from all three "sides".
Three Step Solving Strategy:
Step 1: Simplify each side, if needed.
This would involve things like removing ( ), removing fractions, adding like terms, etc.
Step 2: Use Add./Sub.
Properties to move the variable term on one side and all other terms to the other side.
Step 3: Use Mult./Div.
Properties to remove any values that are in front of the variable.
Example:
\(2(y + 1) <= y - 4\)
or \(2y + 2 <= y - 4\) Step 1
or \(2y + 2 -y <= y - 4 -y\) Step 2
or \(y + 2 <= - 4\)
or \(y + 2 - 2 <= - 4 - 2\) Step 2
or \(y <= - 6\)
Absolute value equations and inequalities
An absolute value of a number \(a\) is represented as, \(|a|\). An absolute equation, \(|x| = a\) consists of two solutions, \(x = a\) and \(x = -a\). Hence, care should be taken while solving an equation consisting of an absolute value.
For example, the equation, \(|x - 17| = 45\) is solved in two different parts.
Part I: \(x - 17 = 45\) making \(x = 62\).
Part II: \(x - 17 = -45\) making \(x = -28\).
An inequality equation with an absolute value, \(|x| > a\) means \(x > a\) and \(x < -a\).
For example,
\(5|3x + 18| < 45\)
In order to solve this equation we first divide it by 5, \(|3x + 18| < 9\).
Now comes the most important step of removing absolute sign and rewriting the equation as, \(-9 < 3x + 18 < 9\).
This equation is finally solved as, \(-27 < 3x < -9\). Hence, \(-9 < x < 3\) is the solution.
gmatclubot
Math Lesson- Solving Linear Inequalities [#permalink]
29 Sep 2016, 12:16
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