How do you know if a value is a solution for an inequality?

How is this different from determining if a value is a solution to an equation? If you replace the equal sign of an equation with an inequality sign, is there ever a time when the same value will be a solution to both the equation and the inequality? Can you show me an example of an inequality and provide a value that may or may not be a solution to the inequality.





How do you know if a value is a solution for an inequality?
You're asking four questions. I'll take them in order.





1) A value is a solution for the inequality if you plug in the value for the variable and the inequality is true. Ie, if X%26gt;5, then X=6 is a solution.





2) The difference is as follows: equations and inequalities both have solution sets - values that will make the equation or inequality true. The difference between 5x=10 and 5x %26gt; 10 is that x=2 satisfies the equation but not the inequality (since 10 is not greater than 10), while anything greater than 2 (including 2.00000001) will satisfy the inequality but not the equation.





3) Yes, the same value will solve both an equation and an inequality if and only if the solution sets overlap. This basically means that you need a ';less than or equal to'; or ';greater than or equal to'; sign.





4) An example of an inequality would be:





4x + 5 %26lt; 17





Try the values 2, 3, and 4. One of these is a solution. The others are not.How do you know if a value is a solution for an inequality?
Simply put, a value is a solution to inequality only if it SATISFIES the inequality. The same is true for an equation; the value is a solution to the equation if it SATISFIES the equation. The only difference between the two is that one is an inequality (less than, greater than, etc.), and the other is an equality (equals sign).





Example:


Determine if x = 6 is a solution to the inequality 4x %26lt; 13.





Plug x = 6 into the inequality and simplify:


4(6) %26lt; 13


24 %26lt; 13





Obviously, 24 is NOT less than 13, so x = 6 is not a solution to the inequality. Now, let's try x = 2 to see if that is a solution. Follow the same process:





2(4) %26lt; 13


8 %26lt; 13





8 is less than 13, so we would say that x = 2 is a solution to the inequality.





Now, there do exist values that will satisfy both an equation and an inequality. Let's take a look at an example:





6x + 2 %26gt; 9


5x - 4 = 11





Let's see if x = 3 will satisfy both the inequality and equality. We'll do the inequality first:





6(3) + 2 %26gt; 9


18 + 2 %26gt; 9


20 %26gt; 9 ...x = 3 satisfies the inequality since 20 is greater than 9





Now do the equality:





5(3) - 4 = 11


15 - 4 = 11


11 = 11 ...x = 3 satisfies the equality since 11 equals 11





I hope this clarifies things for you!