The compound inequality a < x < b means "a is less than x,...

The compound inequality $a < x < b$ means "a is less than $x,$ and $x$ is less than $b$ ". Which one of the following inequalities is not satisfied by some real number $x ?$
A. $-3 < x < 5$
B. $0 < x < 4$
C. $-3 < x < -2$
D. $-7 < x < -10$

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Key Concepts

Ordering Relationships

Ordering relationships, defined by the symbols ‘<’ and ‘>’, establish expected positions on the number line. For a compound inequality to have solutions, the lower bound must indeed be less than the upper bound; if this condition is not met, such as when the lower bound is greater than the upper bound, no real number can satisfy the inequality.

Solution Set

The solution set of a compound inequality is the collection of all values that satisfy every condition stipulated by the inequality. It is important to check that the range defined by the inequality is not empty, which occurs when the lower bound is less than the upper bound, ensuring that there is at least one real number that satisfies the inequality.

Compound Inequalities

A compound inequality involves two inequality statements connected by a logical operator (usually “and”) that must be simultaneously true. This means that the variable must satisfy both parts of the inequality at the same time, and solving such an inequality typically involves finding the intersection of the ranges represented by each part.

Transcript

00:03 For this problem, we're reminded that if we see an expression like a is less than x is less than b, we're to interpret that to mean that a is less than x and x is less than b.

00:15 So both have to be true.

00:17 So in one of these answer choices, a through d, there are no real numbers that satisfy it.

00:24 And for the other three, there are real numbers that satisfy them.

00:28 So i'm going to use a graphical approach to figure out which ones do have solutions and which one does not.

00:35 So let's look at a.

00:37 So we have negative 3 is less than x is less than 5.

00:40 So that means that negative 3 is less than x.

00:44 And another way to write that would be x is greater than negative 3 and x is less than 5.

00:51 So let's graph both of these situations.

00:54 X is greater than negative 3 would look like this.

00:59 X is less than 5 would look like this.

01:02 So there is overlap there.