document.write( "Question 118193: is there more than one way to solve inequalities, perhaps a shortcut?
\n" ); document.write( "can you show me using an example? \n" ); document.write( "

Algebra.Com's Answer #86471 by MathLover1(20850) $\"\"$ $\"About$

You can put this solution on YOUR website!
Solving linear inequalities is $\"very\"$ $\"+similar\"$ to solving $\"linear\"$ $\"+equations\"$ , except for one detail: you $\"flip\"$ $\"+the\"$ $\"+inequality+\"$ $\"sign\"$ whenever you multiply or divide the inequality by a $\"negative\"$ .
\n" ); document.write( "Use the addition, subtraction, multiplication, and division properties of inequalities to solve linear inequalities. We always want to get the variable on one side and everything else on the other side by using inverse operations.
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\n" ); document.write( "some examples:\r
\n" ); document.write( "\n" ); document.write( " $\"x%2B3%3C2\"$ Еadd $\"-3\"$ to the both sides\r
\n" ); document.write( "\n" ); document.write( " $\"x%2B3-3%3C2-3\"$ Е\r
\n" ); document.write( "\n" ); document.write( " $\"x%3C-1\"$ Е\r
\n" ); document.write( "\n" ); document.write( "Solution are all $\"x\"$ values from $\"-infinity\"$ to $\"-1\"$ , where $\"-1\"$ is excluded.
\n" ); document.write( "We can write it like this: interval ( $\"-infinity\"$ , $\"-1\"$ )\r
\n" ); document.write( "\n" ); document.write( "As you can see from this example, the only difference here is that you have a \" $\"less\"$ $\"+than\"$ \" sign, instead of an \" $\"equals\"$ \" sign.
\n" ); document.write( "Note that the solution to a \"less than, but not equal to\" inequality is graphed $\"with\"$ a $\"parentheses\"$ (or else an open dot) at the endpoint.\r
\n" ); document.write( "\n" ); document.write( " $\"4x%2B6%3C=+3x+%96+5\"$ ЕЕ..move $\"3x\"$ to the left and $\"6\"$ to the right\r
\n" ); document.write( "\n" ); document.write( " $\"4x+%96+3x+%3C=+-6+%96+5\"$ ЕЕ..\r
\n" ); document.write( "\n" ); document.write( " $\"x+%3C=+-11\"$ ЕЕ..\r
\n" ); document.write( "\n" ); document.write( "Solution are all $\"x\"$ values from $\"-infinity\"$ to $\"-11\"$ , where $\"-1\"$ is included.
\n" ); document.write( "We can write it like this: interval ( $\"-infinity\"$ , $\"-1\"$ ]
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\n" ); document.write( "Note that the solution to a \"less than or equal to\" inequality is graphed with a bracket (or else a closed dot) at the endpoint.\r
\n" ); document.write( "\n" ); document.write( " $\"-2x+%2B+5+%3E=+-3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"-2x+%3E=+-3+-5\"$ ......divide both sides by $\"-1\"$ (remember, you flip the sign )\r
\n" ); document.write( "\n" ); document.write( " $\"2x+%3C=+3%2B5\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"2x+%3C=+8\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+%3C=+4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Graphing Inequalities\r
\n" ); document.write( "\n" ); document.write( "When $\"x\"$ is less than a constant (for example $\"x%3C5\"$ ), you darken in the part of the number line that is $\"to\"$ $\"+the\"$ $\"+left\"$ of the $\"constant\"$ . \r
\n" ); document.write( "\n" ); document.write( "Also, because there is $\"no\"$ $\"+equal\"$ $\"+line\"$ , we are $\"not\"$ $\"+including\"$ where $\"x\"$ is equal to the constant. \r
\n" ); document.write( "\n" ); document.write( "That means we are not including the $\"endpoint\"$ . One way to notate that is to use an open hole at that point. \r
\n" ); document.write( "\n" ); document.write( "When $\"x\"$ is greater than a constant (for example $\"x%3E5\"$ ), you darken in the part of the number line that is to the right of the constant.
\n" ); document.write( "
\n" ); document.write( "Also, because there is $\"no\"$ $\"+equal\"$ $\"+line\"$ , we are $\"not\"$ $\"+including\"$ where $\"x\"$ is equal to the constant. \r
\n" ); document.write( "\n" ); document.write( "That means we are not including the endpoint. One way to notate that is to use an open hole at that point. \r
\n" ); document.write( "\n" ); document.write( "When $\"x\"$ is less than or equal to a constant (for example $\"x%3C=5\"$ ), you darken in the part of the number line that is to the left of the constant. \r
\n" ); document.write( "\n" ); document.write( "Also, because there is an equal line, we are $\"+including\"$ where $\"x\"$ is equal to the constant. That means we are $\"including\"$ the endpoint. One way to notate that is to use an closed hole at that point. \r
\n" ); document.write( "\n" ); document.write( "When $\"x\"$ is greater than or equal to a constant (for example $\"x%3E=5\"$ ), you darken in the part of the number line that is to the right of the constant. \r
\n" ); document.write( "\n" ); document.write( "Also, because there is an equal line, we are $\"+including\"$ where $\"x\"$ is equal to the constant. That means we are $\"including\"$ the endpoint. One way to notate that is to use an closed hole at that point. \r
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\n" ); document.write( "\n" ); document.write( "Properties for Inequalities: \r
\n" ); document.write( "\n" ); document.write( " Addition/Subtraction Property for Inequalities \r
\n" ); document.write( "\n" ); document.write( "If $\"a+%3C+b\"$ , then $\"+a+%2B+c+%3C+b+%2B+c\"$ \r
\n" ); document.write( "\n" ); document.write( "If $\"a+%3C+b\"$ , then $\"a+-+c+%3C+b+%96+c\"$ \r
\n" ); document.write( "\n" ); document.write( "Multiplication/Division Properties for Inequalities \r
\n" ); document.write( "\n" ); document.write( "when multiplying/dividing by a positive value \r
\n" ); document.write( "\n" ); document.write( "If $\"a+%3C+b\"$ AND $\"+c\"$ is positive, then $\"ac+%3C+bc+\"$ \r
\n" ); document.write( "\n" ); document.write( "If $\"a+%3C+b\"$ AND $\"+c\"$ is positive, then $\"+a%2Fc+%3C+b%2Fc\"$ \r
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