Lesson Video: One-Step Inequalities: Addition or Subtraction Mathematics • 6th Grade

In this video, we will learn how to solve one-step linear inequalities by addition or subtraction.

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Video Transcript

In this video, we will learn how to solve one-step linear inequalities using addition or subtraction. An inequality is a mathematical sentence that contains one or more of the following inequality symbols. There are four inequality symbols, and they show that the value of one expression is greater than the other. On the first row, we have a greater-than symbol, which would say ๐‘ฅ is greater than ๐‘. The second row would say ๐‘ฅ is less than ๐‘. The third row is a slight variation of the first row. This symbol tells us that ๐‘ฅ is greater than or equal to ๐‘. And in a similar way, the fourth row says ๐‘ฅ is less than or equal to ๐‘.

Mathematical sentences that have an equal sign are called equations. This equation says three plus two is equal to five. In equations, both sides of the equal sign have the same value. In an inequality, one side is greater than the other. For example, here we have three plus two is greater than four. When working with inequalities, weโ€™ll often need to solve them. And this means to find a range of values that a variable can take to make the inequality true. If we look at the inequality ๐‘ฅ plus one is greater than 10, then, to solve this inequality, we would need to find all of the values that make ๐‘ฅ plus one greater than 10. And thatโ€™s what weโ€™re going to learn how to do in this video.

Before we look at solving for inequalities, letโ€™s quickly remember the balance method when weโ€™re working with equations. An equation states the equality between two expressions. We can represent it with a balance scale. In this case, we have ๐‘ฅ plus five equals eight. If we want to find just ๐‘ฅ, we can take the five block away. But in order to maintain a balance, we would need to take five away from the other side as well. If we break up the original eight that was on the scale, weโ€™ll have a three and a five. Then weโ€™re able to take that five block away to see that ๐‘ฅ equals three. Algebraically, we represent that by subtracting five from both sides of our equation.

Now, letโ€™s look at how this applies to inequalities. In this case, we want to say that ๐‘ฅ plus five is greater than eight. On the scale, that means the ๐‘ฅ plus five side will weight more; it will be further down than the eight. In this case, when weโ€™re solving for ๐‘ฅ, weโ€™ll need to maintain this uneven scale. The ๐‘ฅ side should remain lower than the eight side. If we try to remove something from the heavier side, in order to keep the proportions the same, weโ€™ll have to remove the same amount from the lighter side.

And just like we did previously, we can break that eight block into two pieces, a three and a five. And when we remove that five block, we see a new inequality emerging. We see the statement that ๐‘ฅ must be greater than three. If we were to write this algebraically, we would subtract five from both sides of the inequality, which leave us with ๐‘ฅ is greater than three. This balance method illustrates a rule for inequalities. And the rule says this: An inequality still holds true when the same number is added to or subtracted from both of its sides. Letโ€™s look at some examples to see how we apply this rule to simplify inequalities.

Given that ๐‘ฅ plus 34 is greater than or equal to 46, what number do you need to add to both sides of the inequality to solve for ๐‘ฅ?

In order to solve for ๐‘ฅ, we want ๐‘ฅ to be the only thing on one side of the inequality. In this case, 34 is being added to ๐‘ฅ. If we wanted to get ๐‘ฅ by itself, we might subtract 34 from both sides of the inequality. And this is where we need to be really careful reading this question because this question wants us to add something to both sides of the equation. This first option, we have subtracted 34. But if our question wants us to do it in a format of addition, then we need to say we are adding negative 34. Since the question has asked what number do we need to add to both sides of the inequality, the answer is we add negative 34. And by adding negative 32 [34] to both sides of the inequality, we find that ๐‘ฅ must be greater than or equal to 12.

Hereโ€™s another example.

If ๐‘ฅ minus eight is greater than negative three, then ๐‘ฅ is greater than blank.

We know that ๐‘ฅ minus eight is greater than negative three. And we want to know then what value must ๐‘ฅ be greater than. And to do that, weโ€™ll want to simplify. Weโ€™ll want to get this ๐‘ฅ by itself on the left side of the greater-than sign. We know that inequalities remain true as long as we add or subtract the same amount from either side of the inequality. I know that ๐‘ฅ minus eight plus eight just equals ๐‘ฅ. And that means we can add eight to both sides of this inequality without changing its value. On the left, we have ๐‘ฅ and, on the right, negative three plus eight equals five. And so we can say if ๐‘ฅ minus eight is greater than negative three, then ๐‘ฅ must be greater than five.

Hereโ€™s a third example.

If ๐‘Ž plus 46 is greater than or equal to 39, then blank. (A) ๐‘Ž is greater than or equal to seven. (B) ๐‘Ž is greater than or equal to negative seven. (C) ๐‘Ž is greater than or equal to 85. (D) ๐‘Ž is greater than or equal to negative 85. Or (E) ๐‘Ž is less than or equal to negative seven.

Weโ€™re starting out with the inequality ๐‘Ž plus 46 is greater than or equal to 39. And we need to simplify it so that we only have an ๐‘Ž term to the left of our inequality symbol. And we know when working with inequalities, as long as we add or subtract the same value from both sides, the inequality remains true. If we want to isolate ๐‘Ž, we can subtract 46 from the left side of the inequality. But if we subtract 46 from the left side of the inequality, we need to subtract 46 from the right side of the inequality. On the left, that just leaves us with ๐‘Ž. ๐‘Ž plus 46 minus 46 equals ๐‘Ž.

And here, we need to be really careful. We have 39, but weโ€™re subtracting more than 39. Weโ€™re subtracting 46. Mathematically, that means we subtract 39 from 46, which gives us seven. But then it takes the sign of the negative 46. 39 take away 46 is negative seven. And weโ€™ve brought down our inequality symbol, because that remains true, to say that ๐‘Ž must be greater than or equal to negative seven, which in this case is option (B).

In our next example, weโ€™ll have to do a bit of rearranging.

Select the option that is equivalent to ๐‘Ž plus seven is less than zero. (A) ๐‘Ž is greater than negative seven. (B) ๐‘Ž is greater than or equal to negative seven. (C) Negative seven is greater than ๐‘Ž. Or (D) negative seven is greater than or equal to ๐‘Ž. (B) ๐‘Ž is greater than or equal to negative seven. (C) Negative seven is greater than ๐‘Ž. Or (D) negative seven is greater than or equal to ๐‘Ž.

First, we write down what we know. ๐‘Ž plus seven is less than zero. We can start to solve this problem by isolating the ๐‘Ž variable. To do that, we subtract seven from the left side of the inequality. And to maintain this true inequality, that means we have to subtract seven from the right side of the inequality. On the left, ๐‘Ž plus seven minus seven equals ๐‘Ž. And on the right, zero minus seven equals negative seven. Weโ€™ve just found that ๐‘Ž is less than negative seven. Now, at first glance, it doesnโ€™t look like we have ๐‘Ž is less than negative seven as an answer choice. To help us clear that up, letโ€™s consider this inequality on a number line.

Weโ€™re dealing with negative seven, and ๐‘Ž has to be less than that. On a number line, less than negative seven would be more negative values. And that would be an arrow pointing to the left. Weโ€™re saying that ๐‘Ž has to be found to the left of negative seven on a number line. Using that information, letโ€™s go back and visit our answer choices. Option (A) says that ๐‘Ž is greater than negative seven. And thatโ€™s the opposite of whatโ€™s true. Option (B) says that ๐‘Ž is greater than or equal to negative seven. Again, that is not true, since ๐‘Ž has to be less than negative seven. Option (C) is an inequality that says negative seven is greater than ๐‘Ž.

If we look at negative seven on our number line, it is always true that negative seven is greater than ๐‘Ž. Negative seven is greater than ๐‘Ž because ๐‘Ž is less than negative seven, which makes option (C) an equivalent statement to ๐‘Ž is less than negative seven. When we look at option (D), it says that negative seven is greater than or equal to ๐‘Ž. But because the inequalities weโ€™re dealing with are only less than and greater than, they donโ€™t have an equal-to component, which makes option (D) false.

In our next example, weโ€™re given a simplified inequality, and weโ€™ll need to rearrange it.

Complete using less than or equal to, less than, greater than, or greater than or equal to: If ๐‘ is less than or equal to negative five, then ๐‘ plus one blank negative four.

We know that ๐‘ is less than or equal to negative five. And we want to know what the relationship of ๐‘ plus one would be to negative four. We know about inequalities that if we add the same number to both sides of an inequality, it will still hold true. If we try to add one to both sides of the inequality, on the left weโ€™ll end up with ๐‘ plus one. We canโ€™t simplify that any further. And on the right, weโ€™ll end up with negative five plus one. We have not changed the value of our inequality. So we have ๐‘ plus one is less than or equal to negative five plus one. And we know that negative five plus one is negative four. And so we confirm that ๐‘ plus one must be less than or equal to negative four.

In our final example, weโ€™ll need to first write an inequality before we solve it.

Daniel wants to buy groceries. He wants to have at least 80 dollars cash with him. And he currently has 15 dollars. Write and solve an inequality that would determine how much cash Daniel should withdraw from the bank.

Daniel wants at least 80 dollars cash. To find out what kind of inequality we should write, letโ€™s think about this. Would it be okay if Daniel had more than 80 dollars cash? Yes. Would it be okay if Daniel had exactly 80 dollars cash? Yes. It would not be okay if he had less than 80 dollars cash. And that means we need an inequality symbol that represents more or equal. We need the money Daniel has to be greater than or equal to 80 dollars. And the money Daniel has consists of two pieces. The money he should withdraw and the 15 dollars he already has. Together, these two values should be greater than or equal to 80 dollars. We can let the variable ๐‘ฅ represent the money that Daniel will withdraw. And so weโ€™re saying that ๐‘ฅ plus 15 must be greater than or equal to 80.

At this point, we have successfully written an inequality to show the money that Daniel needs. But we need to continue by solving this inequality. We want to solve for what values of ๐‘ฅ would this statement be true. And we do that by subtracting 15 dollars from both sides of the inequality. 80 dollars minus 15 dollars equals 65 dollars. ๐‘ฅ must be greater than or equal to 65. And our ๐‘ฅ represents the money that he should withdraw from the bank. Daniel must withdraw at least 65 dollars if he wants to have 80 dollars cash with him. And so an inequality that weโ€™ve written ๐‘ฅ plus 15 is greater than or equal to 80 and solved for ๐‘ฅ must then be greater than or equal to 65.

Weโ€™re now ready to summarize what weโ€™ve learned. An inequality is a mathematical sentence containing an inequality symbol. It shows that the value of one expression is greater than that of the other. To simplify inequalities, we can use the addition and subtraction rule for inequalities, which tells us that an inequality still holds true when the same number is added to or subtracted from both of its sides. Just note that it says equality here, where it should say inequality. An inequality still holds true when the same number is added to or subtracted from both of its sides. And so, if ๐‘ฅ plus three is greater than six, we can subtract three from both sides of this inequality and say that ๐‘ฅ is greater than three.

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