Algebra
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Boundless Algebra
Introduction to Equations, Inequalities, and Graphing
Inequalities
Algebra Textbooks Boundless Algebra Introduction to Equations, Inequalities, and Graphing Inequalities
Algebra Textbooks Boundless Algebra Introduction to Equations, Inequalities, and Graphing
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 9
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Compound Inequalities

A compound inequality involves three expressions, not two, but can also be solved to find the possible values for a variable.

Learning Objective

  • Solve a compound inequality by balancing all three components of the inequality


Key Points

    • A compound inequality is of the following form: $a < x < b$.
    • There are two statements in a compound inequality. The first statement is $a < x$. The next statement is $x < b$. When we read this statement, we say "$a$ is less than $x$, and $x$ is less than $b$."
    • An example of a compound inequality is: $4 < x < 9$. In other words, $x$ is some number strictly between 4 and 9.
    • A compound inequality may contain an expression, such as $1 < x - 6 < 8$; such inequalities can be solved for all possible values of $x$.

Terms

  • inequality

    A statement that of two quantities one is specifically less than or greater than another.

  • compound inequality

    An inequality that is made up of two other inequalities, in the form $a < x < b$.


Full Text

Defining Compound Inequalities

A compound inequality is of the following form:

$a < x < b$

There are actually two statements here. The first statement is $a < x$. The next statement is $x < b$. This statement is therefore read as "$a$ is less than $x$, and $x$ is less than $b$."

The compound inequality $a < x < b$ indicates "betweenness"—the number $x$ is between the numbers $a$ and $b$. Without changing the meaning, the statement $a<x$ can also be read as $x>a$. Therefore, the form $a < x < b$ can also be read as "$x$ is greater than $a$, and at the same time is less than $b$." 

Consider $4 < x < 9$. This states that $x$ is some number strictly between 4 and 9. For a visualization of this inequality, refer to the number line below. The numbers 4 and 9 are not included, so we place open circles on these points.

$4 < x < 9$

The above inequality on the number line.

Similarly, consider $-2 < z < 0$. In this case, $z$ is some number strictly between -2 and 0. Again, because the numbers -2 and 0 are not included, we place open circles on those points. 

$-2 < x < 0$

The above inequality on the number line.

$$Solving Compound Inequalities

Now consider $1 < x + 6 < 8$. The expression $x + 6$ represents some number strictly between 1 and 8. However, the meaning of this is difficult to visualize—what does it mean to say that an expression, rather than a number, lies between two points? Not to worry—we can still find all possible values of not only the expression, but the variable $x$ itself.

The statement $1 < x + 6 < 8$ says that the quantity $x + 6$ is between 1 and 8, a statement that will be true for only certain values of $x$. 

To solve for possible values of $x$, we need to get $x$ by itself:

$1 - 6 < x + 6 - 6 < 8 - 6$

$-5 < x < 2$

Therefore, we find that if $x$ is any number strictly between -5 and 2, the statement $1 < x + 6 < 8$ will be true.

Example 1

Solve $-3 < \dfrac{-2x-7}{5} < 7$.

Multiply each part to remove the denominator from the middle expression: 

$-3\cdot (5) < \dfrac{-2x-7}{5} \cdot (5) < 7 \cdot (5)$

$-15 < -2x-7 < 35$

Isolate $x$ in the middle of the inequality:

$- 15 + 7 < -2x -7 + 7 < 35 + 7$

$- 8 < -2x < 42$

Now divide each part by -2 (and remember to change the direction of the inequality symbol!):

$\displaystyle \frac{-8}{-2} > \frac{-2x}{-2} > \frac{42}{-2}$

$4 > x > -21 $

Finally, it is customary (though not necessary) to write the inequality so that the inequality arrows point to the left (i.e., so that the numbers proceed from smallest to largest):

$-21 < x < 4$

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