Which of the following statements are true? A. If $x \neq a$, then $| x - a | 0$, then $x \neq a$. C. $x$ is a solution of $0

Question

Which of the following statements are true?
A. If

x \neq a

, then

| x - a | < 0

.
B. If

| x - a | > 0

, then

x \neq a

.
C.

x

is a solution of

0 < | x - a | < \delta

if, and only if,

a - \delta < x < a + \delta

.
D.

\lim ( 3 x - 1 ) = 5

means that for all

\delta > 0

, there is some

\varepsilon > 0

such that if

0 < | x - 5 | < \hat { \delta }

, then

\left| \left( \begin{array} { c } x \rightarrow 2 \\ 3 x - 1 \end{array} \right) - 2 \right| < \varepsilon

.

Quizplus · Accepted Answer

- A is false because $| x - a | < 0$ is never true; the absolute value is always non-negative.- B is true because if $| x - a | > 0$, it means $x$ and $a$ are not equal; the absolute value is zero if and only if $x = a$.- C is true because the inequality $0 < | x - a | < \delta$ translates to two cases: $x - a < \delta$ and $- (x - a) < \delta$, which simplify to $a - \delta < x < a + \delta$.- D is false because the limit notation and the conditions for $\delta$ and $\varepsilon$ are incorrectly described.

Which of the Following Statements Are True? A x≠ax \neq ax=a

Which of the Following Statements Are True?
A $x \neq a$