Problem 50 - Algebra 2

Question

Solve each inequality. $$ 2 b^{2}-b<6 $$

Step-by-Step Solution

Answer

The solution is the interval \((-\frac{3}{2}, 2)\).

Step 1: Move All Terms to One Side

Start by moving all the terms to one side of the inequality to form a quadratic inequality. Subtract 6 from both sides of the inequality:\[ 2b^2 - b - 6 < 0 \]

Step 2: Factor the Quadratic Expression

Next, factor the quadratic expression. We are looking for factors of \(-6\) that add up to \(-1\). The equation factors to:\[ (2b + 3)(b - 2) < 0 \]

Step 3: Determine the Critical Points

Find the roots or critical points of the equation by setting each factor equal to zero:- \(2b + 3 = 0\) gives \(b = -\frac{3}{2}\)- \(b - 2 = 0\) gives \(b = 2\)

Step 4: Test Intervals Between Critical Points

Test the intervals around the critical points to determine where the inequality is satisfied:- Interval \((-\infty, -\frac{3}{2})\): Choose \(b = -2\), \((2(-2)+3)((-2)-2) = (-1)(-4) = 4 > 0\)- Interval \((-\frac{3}{2}, 2)\): Choose \(b = 0\), \((2(0)+3)((0)-2) = 3(-2) = -6 < 0\)- Interval \((2, \infty)\): Choose \(b = 3\), \((2(3)+3)((3)-2) = 9 > 0\)Only the interval \((-\frac{3}{2}, 2)\) satisfies the inequality.

Step 5: Write the Solution Interval

Based on the interval testing, the solution to the inequality \(2b^2 - b < 6\) is the set of values that make the expression negative:\((-\frac{3}{2}, 2)\)

Key Concepts

Factoring

In tackling quadratic inequalities, a significant step is factoring the quadratic expression. Factoring involves rewriting the expression as a product of its roots. This simplifies the inequality, making it easier to solve. For example, in the expression \(2b^2 - b - 6 < 0\), we aim to express it in a factored form which is \((2b + 3)(b - 2) < 0\).

This step involves finding two numbers that multiply to give the constant term, which is -6 in this case, and add up to give the linear coefficient, which is -1. The factored form reveals critical points of the inequality or roots of the corresponding equation. This form allows us to determine which intervals may contain feasible solutions.

Critical Points

After factoring a quadratic inequality, it's crucial to determine the critical points. These are values of the variable that make each factor equal to zero. To find them:

• Set each factor to zero.
• Solve for the variable.

In our example, we set \(2b + 3 = 0\) and \(b - 2 = 0\). Solving these gives us the critical points: \(b = -\frac{3}{2}\) and \(b = 2\).

These points divide the number line into intervals. They are vital in determining where the inequality holds true by indicating potential changes in the sign of the expression.

Test Intervals

With critical points known, the next step is to test different intervals they create. Testing intends to find where the inequality is satisfied between these points. The intervals we consider are \((-\infty, -\frac{3}{2})\), \((-\frac{3}{2}, 2)\), and \((2, \infty)\).

For each interval, pick a test point that is not a critical point, substitute it back into the factored inequality, \((2b + 3)(b - 2) < 0\).

• For \((-\infty, -\frac{3}{2})\), choose \(b = -2\), and we find the result is positive.
• For \((-\frac{3}{2}, 2)\), choose \(b = 0\), resulting in a negative value.
• For \((2, \infty)\), choose \(b = 3\), yielding a positive result.

We see that only the interval \((-\frac{3}{2}, 2)\) satisfies the inequality, since it results in a negative value.

Solution Interval

The solution of the inequality is the interval where the inequality holds true. After testing intervals, we concluded that the inequality \(2b^2 - b < 6\) is satisfied in the interval \((-\frac{3}{2}, 2)\). This is the set of values for \(b\) that makes the expression negative, thus fulfilling the condition of the inequality.

This solution interval is determined by analyzing the intervals around the critical points and checking which interval produces results aligned with the inequality's conditions. It summarizes the values for the variable that satisfy the original inequality, giving a concise answer to the problem.