A physicist measures the energies of atoms $A$ and $B$ to be the positive values $a$ and $b$ respectively. He determines that the energy of atom $B$ is between one percent more than and one percent less than twice the energy of atom $A$ . Which of the following systems of inequalities best models the possible range of values for the energies of atoms $A$ and $B$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\left\{\begin{array}{l}a<2.02 b \\ a>1.98 b\end{array}\right.$ $\newline$ (B) $\left\{\begin{array}{l}a<2 b+0.02 \\ a>2 b-0.02\end{array}\right.$ $\newline$ (C) $\left\{\begin{array}{l}b<2.02 a \\ b>1.98 a\end{array}\right.$ $\newline$ (D) $\left\{\begin{array}{l}b<2 a+0.02 \\ b>2 a-0.02\end{array}\right.$

Full solution

Q. A physicist measures the energies of atoms

A

and

B

to be the positive values

a

and

b

respectively. He determines that the energy of atom

B

is between one percent more than and one percent less than twice the energy of atom

A

. Which of the following systems of inequalities best models the possible range of values for the energies of atoms

A

and

B

\newline

Choose

1

answer:

\newline

(A)

\left\{\begin{array}{l}a<2.02 b \\ a>1.98 b\end{array}\right.

\newline

(B)

\left\{\begin{array}{l}a<2 b+0.02 \\ a>2 b-0.02\end{array}\right.

\newline

(C)

\left\{\begin{array}{l}b<2.02 a \\ b>1.98 a\end{array}\right.

\newline

(D)

\left\{\begin{array}{l}b<2 a+0.02 \\ b>2 a-0.02\end{array}\right.

Write Inequality for Atom B's Energy: Atom B's energy is at least $1\%$ more than twice the energy of atom A, so we write the inequality b > 2a \times 1.01.
Simplify First Inequality: Atom B's energy is at most $1\%$ less than twice the energy of atom A, so we write the inequality b < 2a \times 0.99.
Simplify Second Inequality: Simplify the first inequality to b > 2.02a.
Combine Inequalities to Form System: Simplify the second inequality to b < 1.98a.
Combine Inequalities to Form System: Simplify the second inequality to b < 1.98a. Combine the two inequalities to form the system: \{$b > $2$.$02$a, b < $1$.$98$a\}.