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Questions and Answers
What is the first step to solve the inequality $x + 3 < x^2 - 4$?
What is the first step to solve the inequality $x + 3 < x^2 - 4$?
After rearranging the inequality to $x^2 - x - 7 > 0$, what method can be used to find its solutions?
After rearranging the inequality to $x^2 - x - 7 > 0$, what method can be used to find its solutions?
What values of x satisfy the inequality $x^2 - x - 7 > 0$?
What values of x satisfy the inequality $x^2 - x - 7 > 0$?
When graphing the function $y = x^2 - x - 7$, where are the x-intercepts located?
When graphing the function $y = x^2 - x - 7$, where are the x-intercepts located?
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Which of the following describes the graph of the quadratic function related to the inequality $x^2 - x - 7 > 0$?
Which of the following describes the graph of the quadratic function related to the inequality $x^2 - x - 7 > 0$?
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Study Notes
Solving the Inequality
- The first step is to rearrange the terms to get all the terms on one side of the inequality, making it a quadratic expression.
- The inequality can be rewritten as $x^2 - x - 7 > 0$ by subtracting $x+3$ from both sides.
- To find the solutions of the quadratic inequality, the quadratic formula can be applied.
- The quadratic formula provides the roots of the equation $ax^2+bx+c=0$, where a, b, and c are coefficients.
- In this case, a = 1, b = -1, and c = -7.
Finding the Solutions
- The solutions to the inequality $x^2 - x - 7 > 0$ are the values of x that make the expression $x^2 - x - 7$ greater than zero.
- The solutions to $x^2 - x - 7 = 0$ are the values of x where the graph of the function $y = x^2 - x - 7$ intersects the x-axis.
- Solving the equation using the quadratic formula gives the solutions:
- $x = (1 \pm \sqrt{1 + 28}) / 2$
- $x = (1 \pm \sqrt{29}) / 2$
- These are the x-intercepts of the function $y = x^2 - x - 7$.
- Because the inequality is greater than zero, the solutions will be the intervals of the x-axis where the graph is above the x-axis.
- These intervals are $x < (1 - sqrt{29})/2$ or $x > (1 + sqrt{29})/2$.
Graphing the Function
- The graph of the function $y = x^2 - x - 7$ is a parabola that opens upwards.
- The parabola intersects the x-axis at two points which correspond to the solutions of the quadratic equation: $(1 - sqrt{29}) / 2$ and $(1 + sqrt{29}) / 2$.
- The x-intercepts are where the function equals zero.
- The graph of the function $y = x^2 - x - 7$ is a parabola that opens upwards and passes through the x-axis at the two solutions.
- The solution intervals are the areas where the parabola is above the x-axis, meaning y is greater than zero.
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Description
This quiz focuses on solving quadratic inequalities, specifically the inequality x^2 - x - 7 > 0. You'll learn how to rearrange inequalities, find solutions, determine x-intercepts, and describe the graph of the related quadratic function. Test your understanding and improve your skills in quadratic equations and inequalities!
