Problem 96 - Algebra and Trigonometry

Question

Solve and graph the solution set on a number line: \(2 x^{2}+5 x<12\). (Section \(3.6,\) Example 1 )

Step-by-Step Solution

Answer

The solution set is \(-4

Step 1: Rewrite the inequality

Start by rewriting the given inequality so that one side is zero. Subtract 12 from both sides of the equation \((2 x^{2}+5 x-12<0)\).

Step 2: Factor the quadratic expression

Next, factor the quadratic expression on the left side of the equation. The factored form of the equation is \((2x-3)(x+4)<0\).

Step 3: Determine the roots

Set each factor to zero and solve for \(x\) to find the roots. The roots of the equation are \(x=\frac{3}{2}, -4\).

Step 4: Test the intervals

Now, test an \(x\) value from each interval on the number line (\(-\infty, -4\), \(-4, \frac{3}{2}\), \(\frac{3}{2}, \infty)\). This can be done by substituting the value into the inequality and checking whether the inequality holds. Upon testing, the intervals that satisfy the inequality are \(-4

Step 5: Graph the solution set on a number line

On a number line, indicate the solution set \(-4

Key Concepts

Factor Quadratic Expressions

Factoring quadratic expressions is a foundational skill in algebra that allows us to solve quadratic inequalities. A quadratic expression is typically in the form of ax^2 + bx + c, where a, b, and c are constants. The process of factoring involves rewriting this expression as a product of two binomials.

To factor the expression from our example, \(2x^2 + 5x - 12 = 0\), we search for two numbers that multiply to give us \(2 \times -12 = -24\) and add to \(5\). Upon examining possible pairs of factors, we discover that \(8\) and \( -3\) fit the criteria. This gives us the factored form: \[ (2x - 3)(x + 4) < 0 \].

It's crucial to identify the correct pair of numbers that match both multiplication and addition requirements. Practice with various quadratic expressions will help you improve your factoring skills. Remember, not all quadratic expressions can be factored easily, and some may require completing the square or using the quadratic formula.

Graphing on A Number Line

After solving a quadratic inequality, graphing the solution set on a number line visually represents the set of all values that satisfy the inequality. In our example, we have found the critical points or roots to be \(x = -4\) and \(x = \frac{3}{2}\).

When graphing, use a number line with clearly marked points for the roots. Since our inequality is less than zero (\(2x^2 + 5x < 12\)), we are interested in intervals where the quadratic expression is negative.

We graph the solution by shading the interval between \(x = -4\) and \(x = \frac{3}{2}\) and using open circles at these points to indicate they are not included in the solution. This visual representation helps to confirm that we have correctly identified the intervals where the inequality holds true.

Finding the Roots of an Equation

Finding the roots of an equation, also known as solving the equation, is the act of determining the values that make the equation true. In the context of quadratic equations, the roots are the values of \(x\) where the quadratic expression equals zero.

For instance, in our example, we have the quadratic expression \(2x^2 + 5x - 12\). By factoring it into \(2x - 3)(x + 4)\), we can set each factor equal to zero to find the roots: \[ x = \frac{3}{2} \] and \[ x = -4 \].

These roots are critical when solving inequalities because they divide the number line into intervals that we can test to determine where the inequality holds true. It's essential to understand how to find these roots accurately, as they play a significant role in graphing and analyzing the behavior of quadratic functions.

Testing Intervals in Inequalities

When solving quadratic inequalities, testing intervals is a method used to determine which sections of the number line satisfy the inequality.

After factoring the quadratic expression and finding the roots, the number line is divided into intervals by these roots. You test each interval by choosing a sample number from within the interval and substituting it into the original inequality. If the inequality holds true with the sample number, then all numbers within that interval are part of the solution set.

Our original problem, \(2x^2 + 5x < 12\), when factored and tested, revealed that the interval \( -4 < x < \frac{3}{2} \) satisfies the inequality. By meticulously testing each interval, we ensure that we capture all possible solutions to the inequality. Intervals testing is a reliable step in solving quadratic inequalities that requires careful analysis and correct selection of sample numbers to avoid incorrect conclusions.

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Problem 97

Other exercises in this chapter

Problem 96

Solve each equation. $$ 3|\log x|-6=0 $$

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Problem 96

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln e^{13 x} $$

View solution

Problem 97

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s)

View solution

Problem 97

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ e^{\ln 5 x^{2}} $$

View solution

Key Concepts

Factor Quadratic Expressions

Factoring quadratic expressions is a foundational skill in algebra that allows us to solve quadratic inequalities. A quadratic expression is typically in the form of ax^2 + bx + c, where a, b, and c are constants. The process of factoring involves rewriting this expression as a product of two binomials.

To factor the expression from our example, \(2x^2 + 5x - 12 = 0\), we search for two numbers that multiply to give us \(2 \times -12 = -24\) and add to \(5\). Upon examining possible pairs of factors, we discover that \(8\) and \( -3\) fit the criteria. This gives us the factored form: \[ (2x - 3)(x + 4) < 0 \].

It's crucial to identify the correct pair of numbers that match both multiplication and addition requirements. Practice with various quadratic expressions will help you improve your factoring skills. Remember, not all quadratic expressions can be factored easily, and some may require completing the square or using the quadratic formula.

Graphing on A Number Line

After solving a quadratic inequality, graphing the solution set on a number line visually represents the set of all values that satisfy the inequality. In our example, we have found the critical points or roots to be \(x = -4\) and \(x = \frac{3}{2}\).

When graphing, use a number line with clearly marked points for the roots. Since our inequality is less than zero (\(2x^2 + 5x < 12\)), we are interested in intervals where the quadratic expression is negative.

We graph the solution by shading the interval between \(x = -4\) and \(x = \frac{3}{2}\) and using open circles at these points to indicate they are not included in the solution. This visual representation helps to confirm that we have correctly identified the intervals where the inequality holds true.

Finding the Roots of an Equation

Finding the roots of an equation, also known as solving the equation, is the act of determining the values that make the equation true. In the context of quadratic equations, the roots are the values of \(x\) where the quadratic expression equals zero.

For instance, in our example, we have the quadratic expression \(2x^2 + 5x - 12\). By factoring it into \(2x - 3)(x + 4)\), we can set each factor equal to zero to find the roots: \[ x = \frac{3}{2} \] and \[ x = -4 \].

These roots are critical when solving inequalities because they divide the number line into intervals that we can test to determine where the inequality holds true. It's essential to understand how to find these roots accurately, as they play a significant role in graphing and analyzing the behavior of quadratic functions.

Testing Intervals in Inequalities

When solving quadratic inequalities, testing intervals is a method used to determine which sections of the number line satisfy the inequality.

After factoring the quadratic expression and finding the roots, the number line is divided into intervals by these roots. You test each interval by choosing a sample number from within the interval and substituting it into the original inequality. If the inequality holds true with the sample number, then all numbers within that interval are part of the solution set.

Our original problem, \(2x^2 + 5x < 12\), when factored and tested, revealed that the interval \( -4 < x < \frac{3}{2} \) satisfies the inequality. By meticulously testing each interval, we ensure that we capture all possible solutions to the inequality. Intervals testing is a reliable step in solving quadratic inequalities that requires careful analysis and correct selection of sample numbers to avoid incorrect conclusions.