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Chapter 17: Problem 32

Draw a sketch of the graph of the region in which the points satisfy the given system of inequalities. $$\begin{aligned} &16 x+3 y-12>0\\\ &y>x^{2}-2 x-3\\\ &|2 x-3|<3 \end{aligned}$$

Short Answer

Expert verified
The region is between \(x=0\) and \(x=3\), above the line \(y = -\frac{16}{3}x + 4\) and the parabola \(y = x^2 - 2x - 3\).

Step by step solution

01

Simplify the Absolute Value Inequality

Start with the inequality \( |2x - 3| < 3 \). This represents two separate inequalities: \( 2x - 3 < 3 \) and \( 2x - 3 > -3 \). Solve these inequalities to find the range of \( x \).1. \( 2x - 3 < 3 \) leads to \( 2x < 6 \), hence \( x < 3 \).2. \( 2x - 3 > -3 \) leads to \( 2x > 0 \), hence \( x > 0 \).Therefore, the solution is \( 0 < x < 3 \).
02

Simplify the Linear Inequality

The linear inequality \( 16x + 3y - 12 > 0 \) can be rewritten to find \( y \) in terms of \( x \). Move terms around to isolate \( y \):\[ 3y > -16x + 12 \]Divide each term by 3:\[ y > -\frac{16}{3}x + 4 \]
03

Simplify the Quadratic Inequality

Consider the inequality \( y > x^2 - 2x - 3 \). The expression \( x^2 - 2x - 3 \) can be factored or completed to find critical points. It can be factorized as:\[ x^2 - 2x - 3 = (x-3)(x+1) \]Sketch the parabola defined by \( y = x^2 - 2x - 3 \) as an upside-down U-shape opening upwards with roots at \( x = 3 \) and \( x = -1 \). The inequality indicates the region above the parabola.
04

Graph the Inequalities

Now combine all the simplified inequalities. We have:1. Linear inequality: \( y > -\frac{16}{3}x + 4 \)2. Quadratic inequality: \( y > x^2 - 2x - 3 \)3. Absolute value inequality: \( 0 < x < 3 \)Draw each boundary:- The line \( y = -\frac{16}{3}x + 4 \), shading above it.- The parabola \( y = x^2 - 2x - 3 \), shading above it.- Vertical lines at \( x=0 \) and \( x=3 \), shading between them.
05

Identify Intersection Region

The solution region is the overlapping area that satisfies all parts of the system. It lies above both the line \( y = -\frac{16}{3}x + 4 \) and the parabola \( y = x^2 - 2x - 3 \), and is bounded between \( x = 0 \) and \( x = 3 \). Mark this overlap distinctly on your graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Inequalities
Graphing inequalities is about showing which parts of a graph satisfy an inequality condition. When you graph an inequality, you are not only drawing lines or curves, but also shading regions to show the solution set that makes the inequality true. There are several types of inequalities:
  • Linear inequalities: These involve expressions like \( y > -\frac{16}{3}x + 4 \). The graph of a linear inequality results in a half-plane 鈥� a region on one side of a line, not including the line itself, if the inequality is strict, or including the line for a non-strict inequality.
  • Quadratic inequalities: These involve parabolas, like \( y > x^2 - 2x - 3 \). They result in regions above or below the parabola depending on the inequality sign.
  • Absolute value inequalities: These can result in regions between two vertical lines, such as \( 0 < x < 3 \). In these cases, you're interested in all the \( x \) values within a specific range.
To visualize these, plot the boundary of each inequality first as if it were an equal sign, and then shade the region where the inequality holds true. The intersection of all shaded regions in a system of inequalities represents the solution set.
Absolute Value Inequalities
Absolute value inequalities like \( |2x-3| < 3 \) require a bit of unpacking. The absolute value \(|a|\) gives the distance of \(a\) from zero on the number line, regardless of direction, so it is always positive. When solving, look out for these key ideas:
  • Split the inequality into two separate inequalities: For \(|2x - 3| < 3\), you write \(2x - 3 < 3\) and \(2x - 3 > -3\).
  • Solve each part separately: Solving yields \( 0 < x < 3 \), indicating all solutions for \(x\) are between 0 and 3.
In the context of graphing, this means we only consider values of \(x\) within this range when examining solutions to other inequalities in the system. Absolute value inequalities are essential because they narrow down the possible values of \(x\) that satisfy the entire system of equations.
Quadratic Inequalities
Quadratic inequalities, such as \( y > x^2 - 2x - 3 \), involve quadratic equations which graph as parabolas. Understanding these can seem daunting, but breaking them down helps:
  • Factor the quadratic: The equation \(x^2 - 2x - 3\) factors to \((x-3)(x+1)\), giving roots at \( x=3 \) and \( x=-1 \). These roots are crucial as they indicate where the parabola intersects the x-axis.
  • Determine the direction of the parabola: Since the coefficient of \(x^2\) is positive, the parabola opens upwards, meaning any solution for \(y\) is above this curve.
  • The inequality \(y > x^2 - 2x - 3\) implies we are interested in the region above the parabola and not on it.
When dealing with systems, the solution to the quadratic inequality combines with other inequalities to indicate overall solution areas. For instance, in our exercise, the solution lies above this parabola, also considering other graph components, like lines or additional quadrants.

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