Difference between revisions of "1980 AHSME Problems/Problem 6"
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A positive number <math>x</math> satisfies the inequality <math>\sqrt{x} < 2x</math> if and only if | A positive number <math>x</math> satisfies the inequality <math>\sqrt{x} < 2x</math> if and only if | ||
| − | <math>\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} \x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} \x < 4</math> | + | |
| + | <math>\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} \ x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} \ x < 4</math> | ||
| + | |||
| + | == Solution 1 == | ||
| + | |||
| + | Since both sides of this inequality are positive, we may square both sides to get <math>x < 4x^2</math>. Since <math>x</math> is positive, we can divide both sides by <math>x</math> to get <math>1 < 4x</math>. Dividing both sides by <math>4</math> yields <math>\frac{1}{4} < x</math>, or <math>\boxed{(\textbf{A})\ x > \frac{1}{4}}</math>. | ||
| + | |||
| + | == Solution 2 (Graphing) == | ||
| + | |||
| + | Graphing both <math>\sqrt{x}</math> and <math>2x</math> in Quadrant I of the coordinate plane, we see that the two curves intersect at <math>\left(\frac{1}{4}, \frac{1}{2}\right)</math>, and <math>2x > \sqrt{x}</math> whenever <math>\boxed{(\textbf{A})\ x > \frac{1}{4}}</math>. | ||
| + | |||
| + | == See also == | ||
| + | {{AHSME box|year=1980|num-b=5|num-a=7}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 02:14, 23 July 2025
Problem
A positive number 
satisfies the inequality 
if and only if
Solution 1
Since both sides of this inequality are positive, we may square both sides to get 
. Since is positive, we can divide both sides by to get 
. Dividing both sides by 
yields 
, or 
.
Solution 2 (Graphing)
Graphing both 
and 
in Quadrant I of the coordinate plane, we see that the two curves intersect at 
, and 
whenever .
See also
| 1980 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
