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Difference between revisions of "1976 AHSME Problems/Problem 7"

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If <math>-1 < x < 1</math>, then the first factor is positive, and the second factor is also positive.
 
If <math>-1 < x < 1</math>, then the first factor is positive, and the second factor is also positive.
  
−
If <math>x > 1</math>, the first factor is positive, but the second factor is negative.
+
If <math>x > 1</math>, the first factor is negative, but the second factor is positive.
  
 
Combining this with the rules for signs and multiplication, we find that the expression is positive when <math>x < -1</math> or when <math>-1 < x < 1</math>, so our answer is <math>\boxed{\textbf{(C)}}</math> and we are done.
 
Combining this with the rules for signs and multiplication, we find that the expression is positive when <math>x < -1</math> or when <math>-1 < x < 1</math>, so our answer is <math>\boxed{\textbf{(C)}}</math> and we are done.

Latest revision as of 21:02, 24 September 2020

If $x$ is a real number, then the quantity $(1-|x|)(1+x)$ is positive if and only if

$\textbf{(A) }|x|<1\qquad \textbf{(B) }|x|>1\qquad \textbf{(C) }x<-1\text{ or }-1<x<1\qquad\\ \textbf{(D) }x<1\qquad \textbf{(E) }x<-1$

Solution

We divide our solution into three cases: that of $x < -1$, that of $-1 < x < 1$, and that of $x > 1$. (When $x = -1$ or $x = 1$, the expression is zero, therefore not positive.)

If $x < -1$, then the first factor is negative, and the second factor is also negative.

If $-1 < x < 1$, then the first factor is positive, and the second factor is also positive.

If $x > 1$, the first factor is negative, but the second factor is positive.

Combining this with the rules for signs and multiplication, we find that the expression is positive when $x < -1$ or when $-1 < x < 1$, so our answer is $\boxed{\textbf{(C)}}$ and we are done.

See also

1976 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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