Unlike an equation, which leaves you with one final number as your answer, the solution to an inequality is a range of numbers. For example, if your solution is x>4{\displaystyle x>4}, that means any number greater than 4{\displaystyle 4} is a potential solution. When you check possible solutions to inequalities, you’re trying to determine whether the statement made by the equality is true or untrue. If the statement is untrue, your solution is incorrect. For example, you might end up with 5<4{\displaystyle 5<4}. This is an untrue statement because 5 is not less than 4.
To get the x{\displaystyle x} by itself, you have to divide by −2{\displaystyle -2}. Since you’re dividing by a negative number, you also have to flip the inequality sign: −2x−2<6−2{\displaystyle {\frac {-2x}{-2}}<{\frac {6}{-2}}}. Simplify on both sides of the inequality to get your solution: x<−3{\displaystyle x<-3}
To return to the previous example, the original inequality was −2x>6{\displaystyle -2x>6} and the solution was x<−3{\displaystyle x<-3}. Start with −3{\displaystyle -3}. Since the solution is less than −3{\displaystyle -3}, you should get an equation here—and you do: −2(−3)=6{\displaystyle -2(-3)=6}. So that checks out. Now try any number less than −3{\displaystyle -3}, such as −10{\displaystyle -10}: −2(−10)>6{\displaystyle -2(-10)>6} simplifies to 20>6{\displaystyle 20>6}. In words, you would say “20 is greater than 6,” and that’s a true statement—so your solution checks out. Way to go!
23∗32=66=1{\displaystyle {\frac {2}{3}}*{\frac {3}{2}}={\frac {6}{6}}=1} When you have a whole number, like 5{\displaystyle 5}, think of it as 51{\displaystyle {\frac {5}{1}}}. That makes its reciprocal 15{\displaystyle {\frac {1}{5}}}. Taking the reciprocal is also useful when you’re solving for a variable in the denominator of a fraction. Taking the reciprocal on both sides allows you to get that variable by itself.
Alex and Britt both run a 10k. Alex finishes first, running at 5 km/h, while Britt lopes along at a leisurely 2 km/h. Alex’s speed is greater than Britt’s speed: 5>2{\displaystyle 5>2}. To find out how long it took each of them to run the 10k, divide the distance by the speed: 105{\displaystyle {\frac {10}{5}}} for Alex and 102{\displaystyle {\frac {10}{2}}}. Simplify the fractions and you get 2{\displaystyle 2} and 5{\displaystyle 5}. Your inequality sign has reversed, because Alex finished the 10k in less time than Britt did.
Check your solution by using the end point of −15{\displaystyle -15}: −3(−15)=45{\displaystyle -3(-15)=45}. According to your solution, x{\displaystyle x} is less than −15{\displaystyle -15}, so this shows your solution is correct.
Remember: no need to change the sign here because you never multiplied or divided by a negative number.
Plug in a value to test your solution: −3(1)<9{\displaystyle -3(1)<9} gives you −3<9{\displaystyle -3<9}, which is a true statement!
