Pregunta
Graph the solution set of the following system of inequalities.
Ask by Lynch Wyatt. in the United States
Dec 06,2024
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The solution set is the region where the line
and the parabola
overlap.
Solución
To graph the solution set of the system of inequalities given by:
we will follow these steps:
Step 1: Graph the first inequality
-
Convert to slope-intercept form:
-
Graph the line
: - The y-intercept is
. - The x-intercept can be found by setting
:
- The y-intercept is
-
Draw the line and shade below it (since
).
Step 2: Graph the second inequality
-
Graph the parabola
: - The vertex of the parabola is at
. - The parabola opens upwards.
- The vertex of the parabola is at
-
Draw the parabola and shade above it (since
).
Step 3: Identify the solution set
The solution set of the system of inequalities is the region where the shaded areas from both inequalities overlap.
Summary of the Graph
- The line
is drawn with a solid line (since the inequality is ). - The parabola
is drawn with a solid line (since the inequality is ). - The area below the line and above the parabola represents the solution set.
Final Note
To visualize this, you would typically use graphing software or graph paper to accurately plot the line and the parabola, ensuring to shade the correct regions. The intersection points of the line and the parabola can also be found to determine the boundaries of the solution set more precisely.
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Bonus Knowledge
To graph the solution set for the system of inequalities, let’s break it down step-by-step:
First, rewrite the inequalities for easier graphing:
-
Inequality 1:
can be rearranged to or . This is a straight line with a y-intercept at and a slope of . Since it is a “less than or equal to” inequality, the line will be solid. -
Inequality 2:
describes a parabola opening upwards with vertex at . For this inequality, the region will be shaded above the parabola, indicating that this is a “greater than or equal to” situation, so the line of the parabola is also solid.
To graph these inequalities:
-
Start by drawing the line for
. Plot the y-intercept (0, 2) and use the slope to find another point (e.g., for , ). Draw a solid line through these points and shade below it. -
Next, sketch the parabola
. Mark the vertex at (0, -3) and a few additional points, like (-1, -2) and (1, -2). Then draw the parabola, making sure to shade the area above it.
Finally, the solution set for the system of inequalities is where the shaded regions overlap. This area represents all the points
that satisfy both inequalities.

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