Problem 34
Question
Solve the inequality algebraically or graphically. $$-2 x^{2}+2 x+3 \geq 0$$
Step-by-Step Solution
Verified Answer
The solution to the inequality is \(x \) in the interval \( (- ∞, 1]\).
1Step 1: Rearrange the inequality
Rearrange the inequality so that all terms are on one side: \[-2x^{2} + 2x + 3 \geq 0\] becomes \[-2x^{2} + 2x + 3 - 0 \geq 0\] which simplifies to \[-2x^{2} + 2x + 3 \geq 0\]
2Step 2: Factor out the quadratic
Factor the quadratic equation \[-2x^{2} + 2x + 3 = -2(x-1)(x - 1.5)\]
3Step 3: Find the roots
Set the factors equal to 0 to find the roots: \(x-1 = 0\) has root \(x = 1\) and \(x - 1.5 = 0\) has root \(x = 1.5\)
4Step 4: Test the intervals
Now we divide the number line into intervals and check where the inequality holds true. The intervals are \((- ∞, 1)\), \((1, 1.5)\), \((1.5, ∞)\). Test a point in each interval: - Choose \(x = 0\) for interval \((- ∞, 1)\). The inequality \(-2(0)² + 2(0) + 3 \geq 0\) gives \(3 \geq 0\) which is true. - Choose \(x = 1.25\) for interval \((1, 1.5)\). The inequality \(-2(1.25)² + 2(1.25) + 3 \geq 0\) gives \(-0.625 \geq 0\) which is not true. - Choose \(x = 2\) for interval \((1.5, ∞)\). The inequality \(-2(2)² + 2(2) + 3 \geq 0\) gives \(-1 \geq 0\) which is not true. This means the inequality is true for \(x\) in the interval \((- ∞, 1)\).
5Step 5: Include roots where needed
Since our original inequality is \(-2x^{2} + 2x + 3 \geq 0\) we need to include the roots in our solution. The root \(x = 1\) lies in the valid interval \((- ∞, 1)\) but \(x = 1.5\) does not. Therefore the solution is \(x \) in the interval \( (- ∞, 1]\).
Key Concepts
Algebraic SolutionGraphical SolutionFinding Roots
Algebraic Solution
The algebraic approach to solving quadratic inequalities, like \[-2x^2 + 2x + 3 \geq 0\], involves several clear steps. Firstly, we aim to rearrange the inequality so that all terms are on one side.
This means that when you have an inequality, everything should align either to the left or right side of the inequality symbol. For this exercise, it simplifies to an equation by setting equality.Next, we need to factor the quadratic expression. Factoring helps us break down complex equations into simpler ones. It leads to solving a polynomial of a degree less than two, which is much easier. By expressing \(-2x^2 + 2x + 3\) in factor form, we get \[-2(x-1)(x-1.5)\].
This representation makes it much simpler to find the solutions when the expression equals zero. Factorization essentially provides the critical points or boundaries where an inequality may change its truth value. Once you factor, the roots of the quadratic equation \(-2(x-1)(x-1.5) = 0\) can be easily found by setting each factor to zero. These roots will be invaluable when determining the intervals where the inequality holds true.
This means that when you have an inequality, everything should align either to the left or right side of the inequality symbol. For this exercise, it simplifies to an equation by setting equality.Next, we need to factor the quadratic expression. Factoring helps us break down complex equations into simpler ones. It leads to solving a polynomial of a degree less than two, which is much easier. By expressing \(-2x^2 + 2x + 3\) in factor form, we get \[-2(x-1)(x-1.5)\].
This representation makes it much simpler to find the solutions when the expression equals zero. Factorization essentially provides the critical points or boundaries where an inequality may change its truth value. Once you factor, the roots of the quadratic equation \(-2(x-1)(x-1.5) = 0\) can be easily found by setting each factor to zero. These roots will be invaluable when determining the intervals where the inequality holds true.
Graphical Solution
For a graphical solution, visualize the quadratic inequality on a graph.We interpret \(-2x^2 + 2x + 3\) as a parabola.
Since the coefficient of \(x^2\) is negative, the parabola opens downwards. This affects how we analyze the inequality.The roots of the equation from the algebraic solution are critical. They are the x-intercepts or points where the parabola intersects the x-axis.- Root 1: \(x = 1\)- Root 2: \(x = 1.5\)These intercepts divide the number line into distinct intervals, where the curve is above or below the x-axis.Plotting these roots helps us visualize the intervals where the inequality holds true. The parabola will cross the x-axis at these points and between them, you can determine if the inequality \(-2x^2 + 2x + 3 \geq 0\) holds by picking a test point. This is often done visually by analyzing the area above the x-axis as positive, which in the context of this problem indicates true intervals.
Since the coefficient of \(x^2\) is negative, the parabola opens downwards. This affects how we analyze the inequality.The roots of the equation from the algebraic solution are critical. They are the x-intercepts or points where the parabola intersects the x-axis.- Root 1: \(x = 1\)- Root 2: \(x = 1.5\)These intercepts divide the number line into distinct intervals, where the curve is above or below the x-axis.Plotting these roots helps us visualize the intervals where the inequality holds true. The parabola will cross the x-axis at these points and between them, you can determine if the inequality \(-2x^2 + 2x + 3 \geq 0\) holds by picking a test point. This is often done visually by analyzing the area above the x-axis as positive, which in the context of this problem indicates true intervals.
Finding Roots
Finding roots in quadratic inequalities is a crucial step that guides the remaining parts of the solution. A root of a polynomial is a value of \(x\) that makes the polynomial equal zero.
In the equation, \(-2x^2 + 2x + 3 = 0\), our task is to solve for \(x\) that satisfies the equation.Using the distribution from the factors \(-2(x-1)(x-1.5)\), we simply set each factor to zero:- For \(x-1 = 0\), the root is \(x = 1\)- For \(x-1.5 = 0\), the root is \(x = 1.5\)These roots are essentially the x-coordinates where the curve passes through the x-axis.
They segment the number line into permissible regions where inequalities can be validly tested.Identifying these roots is also necessary when dealing solely with algebraic solutions or interpreting graphically. Roots provide endpoints for the intervals where you need to check the solution set. Importantly, they guide us to incorporate or exclude certain endpoints like in our inequality where \(x = 1\) is included. Understanding this step ensures that the quadratic inequality is accurately solved.
In the equation, \(-2x^2 + 2x + 3 = 0\), our task is to solve for \(x\) that satisfies the equation.Using the distribution from the factors \(-2(x-1)(x-1.5)\), we simply set each factor to zero:- For \(x-1 = 0\), the root is \(x = 1\)- For \(x-1.5 = 0\), the root is \(x = 1.5\)These roots are essentially the x-coordinates where the curve passes through the x-axis.
They segment the number line into permissible regions where inequalities can be validly tested.Identifying these roots is also necessary when dealing solely with algebraic solutions or interpreting graphically. Roots provide endpoints for the intervals where you need to check the solution set. Importantly, they guide us to incorporate or exclude certain endpoints like in our inequality where \(x = 1\) is included. Understanding this step ensures that the quadratic inequality is accurately solved.
Other exercises in this chapter
Problem 33
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