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# Inequalities: Inequalities Involving an Absolute Value - the Variable Case

Is \$\$x\gt 0\$\$? >(1) \$\$|7x| \gt 17x\$\$ >(2) \$\$|x + 7| \gt 0\$\$
Correct. [[snippet]] For __Stat. (1)__, remember that an absolute value is always non-negative. The only scenario in which \$\$|7x| \gt 17x\$\$ is when \$\$x\$\$ is negative: * For \$\$x=0\$\$, we know that \$\$|7x| = 17x\$\$, * For \$\$x\gt 0\$\$, \$\$|7x|\$\$ will be smaller than \$\$17x\$\$ (for example, if \$\$x=1\$\$, then \$\$|7|\$\$ is smaller than \$\$17\$\$). Hence, \$\$|7x| \gt 17x\$\$ only when \$\$x\lt 0\$\$. That's a definite "No," but it still considered sufficient, so **Stat.(1) → S → AD**. For __Stat. (2)__, don't bother with solving according to the number case: an absolute value is always non-negative, so \$\$|x+7|\$\$ will always be greater than 0 regardless of the value of \$\$x\$\$. The sole exception for this is when the absolute value is _equal_ to 0, and that only happens when \$\$x=-7\$\$. So Stat. (2) basically tells you that \$\$x\$\$ can equal anything but \$\$-7\$\$. In other words, \$\$x\$\$ can be both positive and negative. No definite answer, so **Stat.(2) → IS → A**.
Incorrect. [[snippet]] For Stat. (2), don't bother with solving according to the number case: an absolute value is always non-negative, so \$\$|x+7|\$\$ will always be greater than 0 regardless of the value of \$\$x\$\$. The sole exception for this is when the absolute value is _equal_ to 0, and that only happens when \$\$x=-7\$\$. So Stat. (2) basically tells you that \$\$x\$\$ can equal anything but \$\$-7\$\$. In other words, \$\$x\$\$ can be both positive and negative. No definite answer, so **Stat.(2) → IS → ACE**.
Incorrect. [[snippet]] For Stat. (1), remember that an absolute value is always non-negative. The only scenario in which \$\$|7x| \gt 17x\$\$ is when \$\$x\$\$ is negative: * For \$\$x=0\$\$, we know that \$\$|7x| = 17x\$\$, * For \$\$x\gt 0\$\$, \$\$|7x|\$\$ will be smaller than \$\$17x\$\$ (for example, if \$\$x=1\$\$, then \$\$|7|\$\$ is smaller than \$\$17\$\$). Hence, \$\$|7x| \gt 17x\$\$ only when \$\$x\lt 0\$\$. That's a definite "No," but it still considered sufficient, so **Stat.(1) → S → AD**.
Incorrect. [[snippet]] For Stat. (2), don't bother with solving according to the number case: an absolute value is always non-negative, so \$\$|x+7|\$\$ will always be greater than 0 regardless of the value of \$\$x\$\$. The sole exception for this is when the absolute value is _equal_ to 0, and that only happens when \$\$x=-7\$\$. So Stat. (2) basically tells you that \$\$x\$\$ can equal anything but \$\$-7\$\$. In other words, \$\$x\$\$ can be both positive and negative. No definite answer, so **Stat.(2) → IS → ACE**.
Incorrect. [[snippet]] For Stat. (1), remember that an absolute value is always non-negative. The only scenario in which \$\$|7x| \gt 17x\$\$ is when \$\$x\$\$ is negative: * For \$\$x=0\$\$, we know that \$\$|7x| = 17x\$\$, * For \$\$x\gt 0\$\$, \$\$|7x|\$\$ will be smaller than \$\$17x\$\$ (for example, if \$\$x=1\$\$, then \$\$|7|\$\$ is smaller than \$\$17\$\$). Hence, \$\$|7x| \gt 17x\$\$ only when \$\$x\lt 0\$\$. That's a definite "No," but it still considered sufficient, so **Stat.(1) → S → AD**.
Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.