QQuestionMathematics
QuestionMathematics
Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.
y, is greater than, minus, one half, x, minus, 3
y>−
2
1
x−3
y, is greater than, x, plus, 6
y>x+ 6
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Answer
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Step 1:: Graph the first inequality, y > - 1 / 2 * x - 3.
First, rewrite the inequality in slope-intercept form (y = mx + b) to find the slope and y-intercept. y > - 1 / 2 * x - 3.5 The slope is - 1 / 2 and the y-intercept is 3.5. Plot the line with a dashed style since the inequality is strict (y is greater than, not greater than or equal to). To indicate the solution set, shade above the line.
Step 2:: Graph the second inequality, y > x + 6.
Rewrite the inequality in slope-intercept form. y > x + 6 The slope is 1 and the y-intercept is 6. Plot the line with a dashed style and shade above the line to indicate the solution set.
Step 3:: Determine the solution set of the system of inequalities.
The solution set is the region that satisfies both inequalities. In this case, it is the area above both lines.
Step 4:: Find a point in the solution set.
One possible point in the solution set is the point of intersection of the two lines. To find this point, solve the system of equations: - 1 / 2 * x - 3.5 = x + 6 Adding 1 / 2 * x to both sides gives:
Step 5:5 = 3 / 2 * x
Multiplying both sides by 2 / 3 gives: x = 13 / 3 Substitute this value back into either equation to find y: y = 1 / 2 * (13 / 3) - 3.5 y = - 11 / 6 So, the point of intersection is (13 / 3, - 11 / 6).
Final Answer
The solution set of the system of inequalities is the area above both lines. A point in the solution set is (13 / 3, - 11 / 6).
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