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Subject Test Math 1

Question 16 of 32

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SAT Subject Test

What are all values of x for which 4 minus x to the power of 2 end exponent greater or equal than x minus 2 invisible times?

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x greater or equal than negative 3

Correct Answer: 
No

negative 5 less or equal than x less or equal than 0

Correct Answer: 
No

negative 3 less or equal than x less or equal than 2

Correct Answer: 
Yes

x less or equal than negative 3 or x greater or equal than 2

Correct Answer: 
No

negative 2 less or equal than x less or equal than 3

Correct Answer: 
No

The correct answer is C.

One way to determine all values of x for which 4 minus x to the power of 2 end exponent greater or equal than x minus 2 invisible times is first to rewrite the inequality in an equivalent form that compares a factored expression to 0 and then reason about the arithmetic sign of the product of the factors. The inequality 4 minus x to the power of 2 end exponent greater or equal than x minus 2 invisible times is equivalent to 4 minus x to the power of 2 end exponent minus x plus 2 invisible times greater or equal than 0, which in turn is equivalent to negative x to the power of 2 end exponent minus x plus 6 invisible times greater or equal than 0. Since negative x to the power of 2 end exponent minus x plus 6 invisible times factors as left parenthesis negative 1 right parenthesis left parenthesis x to the power of 2 end exponent plus x minus 6 invisible times right parenthesis equals left parenthesis negative 1 right parenthesis left parenthesis x plus 3 right parenthesis left parenthesis x minus 2 right parenthesis, the original inequality is equivalent to left parenthesis negative 1 right parenthesis left parenthesis x plus 3 right parenthesis left parenthesis x minus 2 right parenthesis greater or equal than 0 or left parenthesis x plus 3 right parenthesis left parenthesis x minus 2 right parenthesis less or equal than 0. To solve left parenthesis x plus 3 right parenthesis left parenthesis x minus 2 right parenthesis less or equal than 0, notice that this inequality is satisfied by a value of x precisely when either x equals negative 3, x equals 2 or the product of the factors left parenthesis x plus 3 right parenthesis and left parenthesis x minus 2 right parenthesis is negative; this last condition is true for negative 3 less than x less than 2, as shown in the table below:

The figure presents a 4-column table with 3 rows of data. Column 1 does not have a heading. The heading for column 2 is “x is less than negative 3.” The heading for column 3 is “negative 3 is less than x, which is less than 2.” The heading for column 4 is “x is greater than 2.” Column 1 contains the following row headings: “open parenthesis, x plus 3, close parenthesis,” “open parenthesis, x minus 2, close parenthesis,” and “open parenthesis, x plus 3, close parenthesis, times, open parenthesis, x minus 2, close parenthesis.” The data are as follows.  Row 1, open parenthesis, x plus 3, close parenthesis: x is less than negative 3, Negative; negative 3 is less than x, which is less than 2, Positive; x is greater than 2, Positive.  Row 2, open parenthesis, x minus 2, close parenthesis: x is less than negative 3, Negative; negative 3 is less than x, which is less than 2, Negative; x is greater than 2, Positive.  Row 3, open parenthesis, x plus 3, close parenthesis, times, open parenthesis, x minus 2, close parenthesis: x is less than negative 3, Positive; negative 3 is less than x, which is less than 2, Negative; x is greater than 2, Positive.

Therefore, all values of x for which 4 minus x to the power of 2 end exponent greater or equal than x minus 2 invisible times are described by the extended inequality negative 3 less or equal than x less or equal than 2.

Alternatively, one can use a graphing calculator. Graph the two equations y equals 4 minus x to the power of 2 end exponentand y equals x minus 2.

The figure presents a line and a curve in the x y plane with the origin labeled O. The numbers negative 10 through 10, in increments of 5, are indicated on both axes. The line is labeled “y equals x minus 2,” and the curve is labeled “y equals four minus x squared.” The line begins below the x-axis and to the left of the y-axis and ends above the x-axis and to the right of the y-axis. It passes through the points with coordinates 0 comma negative 2 and 2 comma 0. The curve begins below the x-axis and to the left of the y-axis at the point with approximate coordinates negative 3 point 7 comma negative 10. It curves upward and to the right, intersects the line at the point with coordinates negative 3 comma negative 5, crosses the x-axis at the point with coordinates negative 2 comma 0, and reaches a maximum on the y-axis at the point with coordinates 0 comma 4. It then curves downward and to the right, intersects the line on the x-axis at the point with coordinates 2 comma 0, and ends below the x-axis and to the right of the y-axis at the point with approximate coordinates 3 point 7 comma negative 10.

The values of x for which 4 minus x to the power of 2 end exponent greater or equal than x minus 2 invisible times are the same as the values of x for which the parabolic graph of y equals 4 minus x to the power of 2 end exponent lies above or intersects the line y equals x minus 2. Therefore, all values of x for which 4 minus x to the power of 2 end exponent greater or equal than x minus 2 invisible times are described by the extended inequality negative 3 less or equal than x less or equal than 2.

Question Difficulty: 
medium