Explainer: Determining Whether an Inequality Is True or False

In this explainer, we will learn how to know whether an inequality is true or false for a given value of a variable and to determine numbers that satisfy an inequality.

An inequality shows that two expressions are not (necessarily) equal while an equation states the equality of two expressions. A strict inequality states that one quantity is greater than another. However, if the inequality is not strict, the two values could be equal. We write ๐‘Ž<๐‘ to mean that ๐‘Ž is less than and not equal to ๐‘. Similarly, ๐‘Ž>๐‘ means that ๐‘Ž is greater than and not equal to ๐‘. When the inequality is not strict, we write ๐‘Žโ‰ค๐‘ to mean that ๐‘Ž is less than or equal to ๐‘ and ๐‘Žโ‰ฅ๐‘ to mean ๐‘Ž is greater than or equal to ๐‘.

In contrast to an equation, the set of solutions to an inequality is generally a range of values; the solutions to an inequality are the values of the variable for which the inequality holds true. To determine whether a given value is a solution to an inequality, we simply need to replace the variable in the inequality with this value and check whether the inequality holds with the obtained numbers.

Let us see with the first couple of examples how this works.

Example 1: Finding Whether a Value Satisfies an Inequality

Which one of the following numbers satisfies the inequality ๐‘ฅ>โˆ’6?

  1. โˆ’10
  2. โˆ’11
  3. โˆ’9
  4. โˆ’5

Answer

The inequality ๐‘ฅ>โˆ’6 has for solutions all numbers greater than โˆ’6.

Among โˆ’10, โˆ’11, โˆ’9, and โˆ’5, only โˆ’5 is greater than โˆ’6. Therefore, our answer is option D.

Example 2: Identifying a Value That Does Not Satisfy an Inequality

Which of the following does not satisfy the inequality ๐‘ฅโ‰คโˆ’1?

  1. โˆ’3
  2. โˆ’4
  3. โˆ’1
  4. 2

Answer

The inequality ๐‘ฅโ‰คโˆ’1 has for solutions all numbers less than or equal to โˆ’1.

Among โˆ’3, โˆ’4, โˆ’1, and 2, we see that 2 does not satisfy this inequality. Our answer is, therefore, option D.

We will now look at some slightly more complex examples of inequalities, namely, when we do not have only the variable on one side of the inequality but an expression containing the variable.

Example 3: Determining Whether an Inequality Is True or False

Given that ๐‘š=11, is the inequality 8๐‘šโ‰ค64 true or false?

Answer

We need to check here if 8๐‘š, when ๐‘š=11, is less than or equal to 64. For this, we first replace ๐‘š with 11 in 8๐‘š, 8โ‹…11,

and work out this product. We have 8โ‹…11=88. Then, we check whether 88 is less than or equal to 46. We find that 88 is greater than 64. Therefore, the inequality is false when ๐‘š=11.

Example 4: Determining Whether an Inequality Holds for a Particular Value

Given that ๐‘š=5, is the inequality 40๐‘šโ‰ค26 true or false?

Answer

We need to check here if 40๐‘š, when ๐‘š=5, is less than or equal to 26. For this, we first replace ๐‘š with 5 in 40๐‘š, 405,

and calculate the value of this quotient. We find 405=8.

Then, we check whether 8 is less than or equal to 26. We find that 8 is indeed less than 26. Therefore, the inequality is true when ๐‘š=5.

Example 5: Determining Whether an Inequality Is True or False for a Particular Value

Is the inequality 15ร—๐‘ฆโ‰ค93รท3 true or false if ๐‘ฆ=2?

Answer

Here, we need to check whether 15ร—๐‘ฆ is less than or equal to 93รท3 when ๐‘ฆ=2. First, we replace ๐‘ฆ by 2 in 15ร—๐‘ฆ and work out the value of the product. We find that15ร—2=30.

Then, to check whether it is less than or equal to 93รท3, we need to work out the value of this division. We find that 93รท3=31.

And we know that 30 is indeed less than 30. Therefore, the inequality is true when ๐‘ฆ=2.

Key Points

  1. An inequality is a mathematical sentence stating that the value of one expression (or number) is less or greater than the value of another expression (or number).
  2. The solutions to an inequality are the values of the variable for which the inequality holds true.
  3. The set of solutions to an inequality is generally a range of values.
  4. To determine whether a given value is a solution to an inequality, we simply need to replace the variable in the inequality with this value and check with the obtained numbers whether the inequality is true or not.

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