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Which of the regions on the graph contain solutions to the inequality π¦ is greater than or equal to two π₯ minus four?
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When reading the question, it is important to note that the word βregionsβ is plural, which suggests there is more than one region that satisfies the inequality.
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We can begin this question by considering which line on our graph corresponds to the equation π¦ is equal to two π₯ minus four.
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Any linear equation which corresponds to a straight line graph can be written in the form π¦ equals ππ₯ plus π, where π is the slope or gradient and π is the π¦-intercept.
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Our equation has a slope or gradient of two and a π¦-intercept of negative four.
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This means that our graph must cross the π¦-axis at negative four.
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A positive slope or gradient means that our line must slope up from left to right.
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As the slope is equal to two, for every one unit we move to the right, we must move two units up.
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The line that corresponds to the equation π¦ equals two π₯ minus four is shown in pink.
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As the inequality sign is greater than or equal to, we draw a solid line, whereas had it been strictly greater than or strictly less than, we wouldβve drawn a dashed or broken line.
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As the π¦-values need to be greater than or equal to two π₯ minus four, the region must lie above our line.
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There are four such regions on our figure, regions A, B, C, and D.
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The regions on the graph that contain solutions to the inequality π¦ is greater than or equal to two π₯ minus four are A, B, C, and D.
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We can check this answer by substituting in the coordinates of any points in these regions.
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For example, the point with coordinates negative five, five lies in region A.
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The π₯-value here is negative five, and the π¦-value is five.
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Substituting these values into our inequality, we have five is greater than or equal to two multiplied by negative five minus four.
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Two multiplied by negative five is negative 10, and subtracting four from this gives us negative 14.
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As five is greater than or equal to negative 14, this point does satisfy our inequality.
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We could repeat this process for any other point in the regions A, B, C, and D.