Solving an algebraic inequality consists of determining the values of a variable which satisfy the inequality. These values make up a solution set.
The solution to an inequality must respect certain rules. The rules for transforming inequalities make it possible to obtain simpler, equivalent inequalities — that is to say, inequalities that have the same solution set.
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Adding or subtracting the same number from the two sides of an inequality preserves the direction of the inequality.
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Multiplying or dividing the two sides of an inequality by the same positive number preserves the sign of the inequality.
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Multiplying or dividing the two sides of an inequality by the same negative number reverses the direction of the inequality.
The sign of the inequality must be reversed if multiplying or dividing by a negative number.
Consider |2(x+3x+5)\ge 178.|
Isolate |x| to determine the solution set.
||\begin{align} 2(x+3x+5) &\ge 178 \\ (2 \times x)+(2\times 3x)+(2\times 5) &\ge 178 \\ 2x+6x+10 &\ge 178 \\ 8x+10 &\ge 178 \\ 8x &\ge 168\\ x &\ge 21 \end{align}||
The solution set is |x\ge 21.|
For |-\dfrac{5n+1}{2} > 6.|
Isolate |n| to determine the solution set.
||\begin{align} -\dfrac{5n+1}{2} &> 6 \\ -\dfrac{5n+1}{2} \color{red}{\times 2} &> 6 \color{red}{\times 2} \\ -(5n+1) &> 12 \\ -5n-1 &> 12\\ -5n &> 13 \\ \dfrac{-5n}{\color{red}{-5}} &> \dfrac{13}{\color{red}{-5}} \end{align}||
Here, we must reverse the sign of the inequality since we are dividing by a negative number.
||n < -\dfrac{13}{5}||
The solution set is |n < -\dfrac{13}{5}.|