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Solving an equation is the process that makes it possible to determine the value or values of an unknown that validates the equation.
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Solving inequalities is the process that makes it possible to determine the solution set of all of the possible values of an unknown that validates the inequality.
Solving an equation or an inequality consists of finding all the values of the variable that validate the starting equation. Certain rules must be followed when solving equations and inequalities. It is always possible to check if the answer obtained is true by a simple method of validation.
Each type of function has different features, which change the way to solve them. However, they all respect the general rules for transforming an equation.
The rules for transforming equations make it possible to obtain equivalent equations, or equations that have the same solution(s). The equations change depending on the operations performed. Here are the rules associated with each of the operations.
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Add the same number to each side of the equation.
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Subtract the same number from each side of the equation.
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Multiply each side of the equation by the same number that is not equal to zero.
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Divide each side of the equation by the same number that is not equal to zero.
|2x+3=6| where the solution is |\dfrac{3}{2}|
|2x+3\color{red}{+5}=6\color{red}{+5}|
|2x+8=11| where the solution is |\dfrac{3}{2}|
The two equations are equivalent, since they have the same solution.
|6-8x=1| where the solution is |\dfrac{5}{8}|
|(6-8x)\color{red}{\div2}=1\color{red}{\div2}|
|3-4x= \dfrac{1}{2}| where the solution is |\dfrac{5}{8}|
The two equations are equivalent, since they have the same solution.
Just like solving an equation, solving an inequality must also respect certain rules. The rules for transforming inequalities make it possible to obtain equivalent inequalities, or inequalities with the same solution set.
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Adding or subtracting the same number from the two sides of an inequality preserves its meaning.
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Multiplying or dividing the two members of an inequality by the same positive number preserves the meaning of an inequality.
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Multiplying or dividing the two members of an inequality by the same negative number reverses the meaning of an inequality.
|3a-2\ge-16| where the solution-set is |a\ge-\dfrac{14}{3}|
|\color{blue}{5\times}(3a-2)\ge\color{blue}{5\times}-16|
|15a-10\ge-80| where the solution-set is |a\ge-\dfrac{14}{3}|
The two inequalities are equivalent since they have the same solution set.
|-2x+4\le12| where the solution set is |x\ge-4|
|(-2x+4)\color{blue}{\div-2}\le12\color{blue}{\div-2}|
|x+2\color{red}{\ge}-6| where the solution set is |x\ge-4|
The two inequalities are equivalent, since they have the same solution set.
Equivalent equations have the same solution or solutions.
In order to check if two equations are equivalent, we must verify if the solution of one equation validates the second equation.
Consider the following equations: |3x = 27| and |5x = 45.|
The solution of the first equation is |x = 9,| given that |3\times 9 = 27.|
The solution of the second equation is |x = 9,| given that |5\times 9 = 45.|
Therefore, the two equations are equivalent.
To verify the solution, replace the unknown in the starting equation with the solution found.
The solution of the equation |6-8x=1| is |\dfrac{5}{8}.| To verify the solution found, replace the variable |x| by the found solution. ||\begin{align} 6-8 \left(\color{red}{\dfrac{5}{8}}\right) &= 1\\ 6-5 &= 1 \\ 1&=1 \end{align}||
The equality is verified, confirming that the solution of the equation is |x=\dfrac{5}{8}.|
Solving an inequality is similar to solving an equation, with two exceptions:
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the values which satisfy an inequality form a solution set. So, unlike in an equation, there is more than one solution to an inequality;
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when both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality is reversed.