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Together, two sports clubs (A and B) have
x members
A has a members and
B has
b members. Some of the persons are members of both sports clubs. Which of the following expressions describes how many persons are members in only one of the two sports clubs?
(A)
x+a-b
(B)
2(a+b)-2x
(C)
ab-2x
(D)
2x-(a+b)

Together, two sports clubs (A and B) have $x$ members $A$ has a members and $B$ has $b$ members. Some of the persons are members of both sports clubs. Which of the following expressions describes how many persons are members in only one of the two sports clubs? $\newline$ (A) $x+a-b$ $\newline$ (B) $2(a+b)-2 x$ $\newline$ (C) $a b-2 x$ $\newline$ (D) $2 \mathrm{x}-(\mathrm{a}+\mathrm{b})$

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Write two-variable inequalities: word problems

Full solution

Q. Together, two sports clubs (A and B) have

x

members

A

has a members and

B

has

b

members. Some of the persons are members of both sports clubs. Which of the following expressions describes how many persons are members in only one of the two sports clubs?

\newline

(A)

x+a-b

\newline

(B)

2(a+b)-2 x

\newline

(C)

a b-2 x

\newline

(D)

2 \mathrm{x}-(\mathrm{a}+\mathrm{b})

Denote number of people: Let's denote the number of people who are members of both clubs as $c$ . The total number of members in both clubs without double counting is $a + b - c$ .
Total members in both clubs: The total number of members in both clubs including those who are members of both $x$ is different from the sum of individual club members $a + b$ because it doesn't include double-counted members. So, $x = a + b - c$ .
Number of people in one club: To find the number of people who are members in only one club, we need to subtract the number of people who are members of both clubs $c$ from the total number of members in each club. This gives us $(a - c) + (b - c)$ .
Substitute and simplify: Simplifying the expression from the previous step, we get $a + b - 2c$ . But we know that $x = a + b - c$ , so we can replace $a + b$ with $x + c$ .
Find $x - c$ : Substituting $a + b$ with $x + c$ in the expression $a + b - 2c$ , we get $x + c - 2c$ which simplifies to $x - c$ .
Expression for total members: Now we need to find an expression that represents $x - c$ using the options given. The correct expression must account for the total number of members in both clubs ( $x$ ) and subtract the members who are in both clubs ( $c$ ).
Correct expression: Looking at the options, the expression that represents the total number of members in both clubs minus those who are in both clubs is $2x - (a + b)$ . This is because $2x$ represents double counting the members who are in both clubs, and subtracting $a + b$ removes one count of each club's members, leaving only the members who are in one club.
Correct expression: Looking at the options, the expression that represents the total number of members in both clubs minus those who are in both clubs is $2x - (a + b)$ . This is because $2x$ represents double counting the members who are in both clubs, and subtracting $a + b$ removes one count of each club's members, leaving only the members who are in one club.Therefore, the correct expression is (D) $2x - (a + b)$ .

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