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Checking Your Solution

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[MUSIC PLAYING] Let's look at our objectives for today. We'll start by explaining how to assess the reasonableness of your solution to a problem. We'll then look at how to check your solution when solving an equation. And finally, we'll look at how to check your solution when solving an inequality.

Let's start by looking at how to tell if your solution to a problem is reasonable. When solving math problems, it's a good idea to check to make sure your answer is reasonable or correct. Checking to make sure the answer is reasonable does not necessarily mean that it is correct, but that it makes logical sense.

For example, suppose we want to find 20% of $50. The correct solution is $10. Well, common mistakes could lead you to find solutions of $100 or $1,000. Therefore, a reasonable answer may be $12 or $13 dollars.

Now let's look at how we can verify a solution when solving an equation. Verifying a solution when solving an equation involves plugging the solution back into the equation to get a true statement. Let's do an example.

We want to solve the equation 3 times x plus 2 minus 8 equals negative x plus 6. We start by distributing the 3 on the outside of the parentheses, which gives us 3x plus 6 minus 8 equals negative x plus 6. We then combine 6 minus 8 on the left side to get negative 14. So we have 3x minus 14 equals negative x plus 6.

We add x on both sides, which gives us 4x minus 14 equals 6. We then add 14 on both sides, which gives us 4x equals 20. And finally, dividing by 4 gives us x equals 5. So now that we have a solution, we can substitute 5 back for x into the original equation to see if it's correct.

Doing this gives us 3 times 5 plus 2 minus 8 equals negative 5 plus 6. We begin by simplifying 5 plus 2 in the parentheses, which gives us 7. We multiply by 3 to give us 21. And finally, we subtract 8 to have the simplified answer of 13 on the left side of the equation.

Simplifying on the right side, negative 5 plus 6 gives us a positive 1. So we have 13 equals 1, which is a false statement. Therefore, our solution of x equals 5 is incorrect. So let's review our steps in solving the equation to see where we went wrong.

We started by distributing, which we did correctly, 3 times x is 3x and 3 times 2 is 6. Then we combined like terms, 6 and minus 8, and we see the mistake. 6 minus 8 is negative 2, not 14. So let's correct our mistake. The equation should be 3x minus 2 equals negative x plus 6.

From there, we complete the same steps, adding x to both sides, adding 2 to both sides, and dividing by both sides. Now we have a solution of x equals 2. So let's re-verify our new solution by substituting 2 into our equation for x, which gives us 3 times 2 plus 2 minus 8 equals negative 2 plus 6.

We simplify on the left side, 3 times 4 is 12, and 12 minus 8 is 4. Simplifying on the right side, negative 2 plus 6 is 4. And we have 4 equals 4, which is a true statement. So our answer is correct.

Make sure to be careful, because it's also possible to find a correct answer when solving the equation, but to make a mistake while checking the answer. So it helps to double check before assuming a mistake was made while trying to solve the equation.

Now let's look at an example of verifying a solution when solving an inequality. We want to solve the inequality 5x is less than negative x plus 6. So we begin by distributing the negative to the terms in the parentheses, to give us negative x plus 6. And we have our 5x on the left side.

We then add x to both sides, which gives us 6x is less than 6. And finally, we divide by 6 on both sides to give us x is less than 1. The solution set, x is less than 1, tells us that all values within the set will satisfy the inequality, while other values will not. So let's pick some values to test our solution set.

Any statements that are false but we expect to be true, or any statements that are true but we expect to be false indicate an error. Let's start with x equals negative 10. We expect this to be true, because negative 10 is less than 1. So we expect to have a true statement.

Substituting negative 10 in for x gives us 5 times negative 10 is less than negative negative 10 plus 6. Simplifying on the left gives us negative 50. And simplifying on the right gives us a negative negative 4, which is positive 4. So we have negative 50 is less than 4, which is a true statement.

Let's try another point, x equals 2. We expect this to be false because 2 is not less than 1. So we expect to have a false statement. Substituting 2 in for x in our inequality gives us 5 times 2 is less than negative 2 plus 6. Simplifying on the left gives us 10, and simplifying on the right gives us negative 8. 10 is not less than negative 8, so this is a false statement as we expected.

But we may still have an error, so let's try one other point. Let's try x equals 0. We expect this to yield a true statement, because 0 is less than 1. So substituting this back into our original inequality gives us 5 times 0 is less than negative 0 plus 6. On the left gives us 0. And on the right we have negative 6. So 0 is less than negative 6 is a false statement, leading us to believe there was an error made.

So let's go back and check our original work in solving the inequality. We started with distribution and immediately see our mistake. We needed to distribute the negative to both terms in the parentheses, so the inequality should be 5x is less than negative x minus 6. So now continuing to solve, we still add 6 on both sides to give us 6x is less than negative 6. We then divide by 6 on both sides to give us x is less than negative 1.

So we test this solution set, and choose an x value, x equals negative 2. Substituting negative 2 in the inequality gives us 5 times negative 2 is less than negative negative 2 plus 6. Simplifying on both the left and the right sides gives us negative 10 is less than negative 4, which is a true statement as we had expected.

Let's go over our key points from today. Make sure you get these in your notes so you can refer to them later. When solving math problems, it's a good idea to check to make sure your answer is reasonable or correct. Checking to make sure the answer is reasonable does not necessarily mean that it is correct, but that it makes logical sense.

Verifying a solution when solving an equation or inequality involves plugging the solution back into the equation or inequality to get a true statement. So I hope that these key points and examples helped you understand a little bit more about checking your solution. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.