How to Solve a Simple Linear Inequality

Step-by-Step Guide

1. 1

   Step 1: Understand the inequality signs.

   An inequality is like an equation, except instead of saying that the two values are equal, an inequality shows a “greater than” or “less than” relationship.
   
   The >{\displaystyle >} sign means “greater than.” The <{\displaystyle <} means “less than.”For example, 2x+32>−15+x{\displaystyle 2x+{\frac {3}{2}}>-15+x} means that the value on the left side of the inequality is greater than the value on the right side.
2. 2

   Step 2: Combine like terms

   You can solve inequalities using the same algebraic principles you would use to solve an equation.You might need to combine variables, multiply to cancel out fractions, or use other operations to make the numbers easier to work with.
   
   Remember that you need to keep the inequality balanced, so whatever operation you perform on one side of the inequality, you must also perform on the other side.
   
   For example, if solving the inequality 2x+32>−15+x{\displaystyle 2x+{\frac {3}{2}}>-15+x}, you would first multiply each part by 2 to cancel out the fraction:2(2x+32)>2(−15+x){\displaystyle 2(2x+{\frac {3}{2}})>2(-15+x)}4x+3>−30+2x{\displaystyle 4x+3>-30+2x} , To do this, add or subtract variables from one side of the inequality.
   
   Remember that whatever you do to one side, you must also do to the other side.
   
   For example, in the inequality 4x+3>−30+2x{\displaystyle 4x+3>-30+2x}, to move the variable to one side, you would subtract 2x{\displaystyle 2x} from both sides of the inequality:4x+3>−30+2x{\displaystyle 4x+3>-30+2x}4x+3−2x>−30+2x−2x{\displaystyle 4x+3-2x>-30+2x-2x}2x+3>−30{\displaystyle 2x+3>-30} , In order to solve the inequality, the variable should be on one side, without coefficients or constants.
   
   Divide to cancel out coefficients, and add or subtract to remove constants.
   
   Once you have isolated the variable, you have solved the inequality.
   
   For example, in the inequality 2x+3>−30{\displaystyle 2x+3>-30}, to isolate the x{\displaystyle x} you need to subtract 3 from both sides, then divide both sides by 2:2x+3−3>−30−3{\displaystyle 2x+3-3>-30-3}2x>−33{\displaystyle 2x>-33}2x2>−332{\displaystyle {\frac {2x}{2}}>{\frac {-33}{2}}}x>−1612{\displaystyle x>-16{\frac {1}{2}}}
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   Step 3: or otherwise simplify the inequality.

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   Step 4: Move the variable to one side of the inequality.

5. 5

   Step 5: Isolate the variable.

Detailed Guide

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