Use Sophia to knock out your gen-ed requirements quickly and affordably. Learn more
Become a Member Try for Free
×

Checking Your Solution

Author: Sophia
PDF Version

what's covered
This tutorial covers checking your solution when solving an equation, through the discussion of:

Table of Contents

1. Assessing the Reasonableness of a Solution

When solving math problems, it’s a good idea to check that your answer is reasonable or correct. Checking to make sure the answer is reasonable does not necessarily mean that it is correct, but rather that it makes logical sense.

EXAMPLE

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

2. Checking the Solution to an Equation

You can verify a solution when solving an equation by plugging the solution back into the equation to get a true statement.

EXAMPLE

Evaluate 3 open parentheses x plus 2 close parentheses minus 8 equals short dash x plus 6.

3 open parentheses x plus 2 close parentheses minus 8 equals short dash x plus 6 Our Equation
3 x plus 6 minus 8 equals short dash x plus 6 Start by distributing the 3 on the outside of the parentheses.
3 x minus 2 equals short dash x plus 6 Next, combine like terms, 6 and minus 8, which equals negative 2.
table attributes columnalign left end attributes row cell space space 3 x minus 2 equals short dash x plus 6 end cell row cell plus x space space space space space space space space space space plus x end cell end table From there, add x to both sides.
table attributes columnalign left end attributes row cell 4 x minus 2 equals 6 end cell row cell space space space space plus 2 space space space plus 2 end cell end table Then, add 2 to both sides.
table attributes columnalign left end attributes row cell space 4 x equals 8 end cell row cell stack space 4 space with bar on top space space space stack space 4 space with bar on top end cell end table Finally, divide each side by 4.
x equals 2 Our Solution

Now that we have our solution, we can verify it by substituting 2 in the equation for x:

table attributes columnalign left end attributes row cell 3 open parentheses 2 plus 2 close parentheses minus 8 equals short dash 2 plus 6 end cell row cell 3 open parentheses 4 close parentheses minus 8 equals short dash 2 plus 6 end cell row cell 12 minus 8 equals short dash 2 plus 6 end cell row cell 4 equals 4 end cell end table

However, what happens when your solution is incorrect?

EXAMPLE

What if you solved the equation in the manner shown below, and incorrectly combined 6 minus 8 on the left side to get -14? Ultimately, you’d arrive at an incorrect solution of x equals 5.

table attributes columnalign left end attributes row cell 3 x plus 6 minus 8 equals negative x plus 6 end cell row cell 3 x minus 14 equals negative x plus 6 end cell row cell 3 x plus x minus 14 equals negative x plus x plus 6 end cell row cell 4 x minus 14 equals 6 end cell row cell 4 x minus 14 plus 14 equals 6 plus 14 end cell row cell 4 x equals 20 end cell row cell fraction numerator 4 x over denominator 4 end fraction equals 20 over 4 end cell row cell space space space space space x equals 5 end cell end table

In the same way that you verified the correct solution above, consider what happens when you attempt to verify your incorrect solution of x equals 5, by substituting 5 into your equation for x.

table attributes columnalign left end attributes row cell 3 open parentheses 5 close parentheses plus 6 minus 8 equals short dash open parentheses 5 close parentheses plus 6 end cell row cell 15 plus 6 minus 8 equals short dash 5 plus 6 end cell row cell 13 not equal to 1 end cell end table

Simplify 5 plus 2 in the parentheses, which equals 7. Multiply by 3 to give you 21. Finally, subtract 8 to arrive at the simplified answer of 13 on the left side of the equation. Simplifying on the right side, -5 plus 6, equals 1. Your final equation is 13 equals 1, which is a false statement. Therefore, your solution of x equals 5 is incorrect.

big idea
When you find that your solution is incorrect, you need to review your steps in solving the equation to see where you went wrong. 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. Therefore, it helps to double-check before assuming a mistake was made while trying to solve the equation.

3. Checking the Solution to an Inequality

You can also verify a solution when solving an inequality in the same way you check a solution to an equation.

EXAMPLE

Evaluate 5 x less than short dash open parentheses x plus 6 close parentheses.

5 x less than short dash open parentheses x plus 6 close parentheses Our Inequality
5 x less than short dash x minus 6 Start by distributing the negative to the terms in the parentheses.
table attributes columnalign left end attributes row cell space space 5 x less than short dash x minus 6 end cell row cell plus x space space space space plus x end cell end table Add x on both sides.
table attributes columnalign left end attributes row cell 6 x less than short dash 6 end cell row cell stack space 6 space with bar on top space space space space space stack space space 6 space with bar on top end cell end table Divide both sides by 6.
x less than short dash 1 Our Solution

Next, to test our solution set, we will choose a value for x within our solution set, such as -2, because -2 is less than -1. Substituting negative 2 in the inequality and simplifying on both the left and the right side provides:

table attributes columnalign left end attributes row cell 5 open parentheses short dash 2 close parentheses less than short dash open parentheses short dash 2 plus 6 close parentheses end cell row cell short dash 10 less than short dash 4 end cell end table

As you can see, the final expression is negative 10 is less than negative 4, which is a true statement, so your solution is correct.

However, the same process of solving and verification applies even if your solution is incorrect. Any statements that are false but that you expect to be true, or any statements that are true but that you expect to be false, indicate an error. You can test these statements in the following example by using an incorrect solution.

EXAMPLE

Suppose that during the initial distribution when solving the equation, you neglect to distribute the negative to both terms in the parentheses.

table attributes columnalign left end attributes row cell 5 x less than short dash open parentheses x plus 6 close parentheses end cell row cell 5 x less than short dash x plus 6 end cell row cell 6 x less than 6 end cell row cell x less than 1 end cell end table

Now, the solution set, x is less than 1, tells you that all values within the set will satisfy the inequality, while other values will not. Therefore, you would pick some values to test your solution set. Suppose you start with x equals -10. You expect this to be true, because -10 is less than 1; therefore, you expect to have a true statement.

table attributes columnalign left end attributes row cell table attributes columnalign left end attributes row cell 5 left parenthesis short dash 10 right parenthesis less than short dash left parenthesis short dash 10 plus 6 right parenthesis end cell row cell short dash 50 less than short dash left parenthesis short dash 4 right parenthesis end cell row cell short dash 50 less than 4 end cell end table end cell row cell T R U E end cell end table

Substituting -10 in for x in the equation simplifies to -50 is less than 4, which is a true statement. So far, so good!

Next, try another point, x equals 2. You expect this to be false because 2 is not less than 1, so you expect to have a false statement.

table attributes columnalign left end attributes row cell 5 open parentheses 2 close parentheses less than short dash open parentheses 2 plus 6 close parentheses end cell row cell 10 less than short dash 8 end cell row cell T R U E end cell end table

Substituting 2 in for x in your inequality simplifies to 10 is less than -8, which is a false statement, as you expected.

However, you may still have an error, so try one other point, x equals 0. You expect this to yield a true statement because 0 is less than 1.

table attributes columnalign left end attributes row cell 5 open parentheses 0 close parentheses less than short dash open parentheses 0 plus 6 close parentheses end cell row cell 0 less than short dash 6 end cell row cell F A L S E end cell end table

Your final expression is 0 is less than negative 6, which is a false statement, leading you to believe that there was an error made. In this case, you would then go back and check your original work in solving the inequality.

summary
Today you learned that when solving math problems, it’s a good idea to check your solution. You learned that one step in checking your solution is to assess the reasonableness of your solution, which does not necessarily mean that it is correct, but that it makes logical sense. You also learned that checking a solution when solving an equation or inequality involves plugging the solution back into the equation or inequality to get a true statement.

Source: This work is adapted from Sophia author Colleen Atakpu.

© 2024 SOPHIA Learning, LLC. SOPHIA is a registered trademark of SOPHIA Learning, LLC.