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Algebra / Linear inequalities in one or two variables Difficulty: Easy

The shaded region shown represents the solutions to which inequality?

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Explanation

Choice A is correct. The equation for the line representing the boundary of the shaded region can be written in slope-intercept form $y = m x + b$ , where $m$ is the slope and $left parenthesis 0 comma b right parenthesis$ is the y-intercept of the line. For the graph shown, the boundary line passes through the points $left parenthesis 0 comma negative 6 right parenthesis$ and $left parenthesis 9 comma 0 right parenthesis$ . Given two points on a line, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , the slope of the line can be calculated using the equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Substituting the points $left parenthesis 0 comma negative 6 right parenthesis$ and $left parenthesis 9 comma 0 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in this equation yields $m = \frac{0 - (- 6)}{9 - 0}$ , which is equivalent to $m = \frac{6}{9}$ , or $m = \frac{2}{3}$ . Since the point $left parenthesis 0 comma negative 6 right parenthesis$ represents the y-intercept, it follows that $b = - 6$ . Substituting $\frac{2}{3}$ for $m$ and $negative 6$ for $b$ in the equation $y = m x + b$ yields $y = \frac{2}{3} x - 6$ as the equation of the boundary line. Since the shaded region represents all the points on and above this boundary line, it follows that the shaded region shown represents the solutions to the inequality $y \geq \frac{2}{3} x - 6$ .

Choice B is incorrect. This inequality represents a region whose boundary line has a y-intercept of $left parenthesis 0 comma 6 right parenthesis$ , not $left parenthesis 0 comma negative 6 right parenthesis$ .

Choice C is incorrect. This inequality represents a region whose boundary line has a y-intercept of $left parenthesis 0 comma negative 9 right parenthesis$ , not $left parenthesis 0 comma negative 6 right parenthesis$ .

Choice D is incorrect. This inequality represents a region whose boundary line has a y-intercept of $left parenthesis 0 comma 9 right parenthesis$ , not $left parenthesis 0 comma negative 6 right parenthesis$ .