3 Ways To Solve A Simple Linear Inequality

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.

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}

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}

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}}}

For example, to solve the inequality −5x>20{\displaystyle -5x>20}, you need to divide each side by −5{\displaystyle -5} to isolate the variable. Thus, you need to reverse the direction of the inequality sign:−5x>20{\displaystyle -5x>20}−5x−5>20−5{\displaystyle {\frac {-5x}{-5}}>{\frac {20}{-5}}}x<−4{\displaystyle x<-4}

For example, to solve the inequality 6<1x{\displaystyle 6<{\frac {1}{x}}}, you would isolate the x{\displaystyle x} by taking the reciprocal of both sides. Since both sides are positive, you need to reverse the inequality sign:6<1x{\displaystyle 6<{\frac {1}{x}}}16>x{\displaystyle {\frac {1}{6}}>x}

Combine like terms on the right side of the inequality: 3x+2<6+2x{\displaystyle 3x+2<6+2x} Move the variable to one side by subtracting 2x{\displaystyle 2x} from both sides:3x+2−2x<6+2x−2x{\displaystyle 3x+2-2x<6+2x-2x}x+2<6{\displaystyle x+2<6} Isolate the variable by subtracting 2 from both sides:x+2−2<6−2{\displaystyle x+2-2<6-2}x<4{\displaystyle x<4}

Isolate the variable by adding 18 to both sides:−6x−18+18>12+18{\displaystyle -6x-18+18>12+18}−6x>30{\displaystyle -6x>30} Divide both sides by -6. Since you are dividing by a negative number, you need to reverse the inequality sign:−6x>30{\displaystyle -6x>30}−6x−6>30−6{\displaystyle {\frac {-6x}{-6}}>{\frac {30}{-6}}}x<−5{\displaystyle x<-5}

To isolate the variable, subtract 4 from all three parts:14−4<2x+4−4<22−4{\displaystyle 14-4<2x+4-4<22-4}10<2x<18{\displaystyle 10<2x<18} Divide each part by 2:102<2x2<182{\displaystyle {\frac {10}{2}}<{\frac {2x}{2}}<{\frac {18}{2}}}5<x<9{\displaystyle 5<x<9}