Problem 20 - Precalculus : Building Concepts and Connections

Question

Solve the inequality by factoring. $$6 x^{2}-5 x<6$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < -1/2\) or \(-1/2 < x < 2\).

Step 1: Rewrite the inequality in standard form

Subtract 6 from both sides to set the inequality to zero: \(6x^{2}-5x-6<0\).

Step 2: Factor the quadratic expression

Factorize the quadratic equation into two binomial expressions: \(6x^{2}-5x-6 = (2x+1)(3x-6)\).

Step 3: Find the critical points

Set each factor equal to zero and solve for x to get the critical points: If \(2x+1=0\) then \(x=-\frac{1}{2}\). Also, if \(3x-6=0\) then \(x=2\). These critical points split the number line into three intervals.

Step 4: Test each interval

Pick a number from each interval and substitute x in the original inequality to see if it holds.Test for \(x<-1/2\), choose \(x=-1\). Substituting \(x=-1\) gives \(-6 - 5 < 6\) which is true.For \(-1/2 < x < 2\), choose \(x=0\). Testing gives \(-6 < 6\) which is also true.Finally, for \(x>2\), choose \(x=3\). The test gives \(48 - 15 < 6\) which is false.

Step 5: Write the solution

Based on the tests, the inequality holds for intervals \(x < -1/2\) and \(-1/2 < x < 2\).

Key Concepts

Quadratic Inequalities

When dealing with quadratic inequalities, we are essentially looking to find values for which a quadratic expression becomes either greater than or less than zero. These inequalities involve quadratic functions, which is when you see the highest power of the variable as squared, like in the equation provided, \(6x^2 - 5x - 6 < 0\). To solve any quadratic inequality, our main goal is to determine where the quadratic expression crosses the x-axis. Remember:

• The solutions or roots are not just numbers but continuous intervals.
• The inequality indicates a region, either above or below the x-axis, depending on whether it is 'greater than' or 'less than'.

Understanding this concept sets the stage for solving the inequality efficiently.

Factoring

Factoring is a useful technique for simplifying quadratic inequalities because it allows us to rewrite the expression in a form that is easier to evaluate. The key here is to express the quadratic term in the format \((ax + b)(cx + d)\). For our problem, we factorized \(6x^2 - 5x - 6\) into \((2x + 1)(3x - 6)\). This transformation is critical:

• Factoring breaks down the quadratic equation into linear components.
• The factors help us uncover the values of \(x\) where the expression could be zero.

If factoring seems challenging, practice cases like pairing and rearranging terms or using techniques such as the quadratic formula when necessary. Once it is in factored form, we can determine its roots by setting each factor to zero.

Critical Points

Critical points are essentially the solutions obtained when each factor of the quadratic expression equals zero. These points help divide the number line into distinct intervals, which is essential in solving inequalities. For it, we solve the equations derived from \((2x + 1)(3x - 6) = 0\):

• For \(2x + 1 = 0\), the critical point is \(x = -\frac{1}{2}\).
• For \(3x - 6 = 0\), the critical point is \(x = 2\).

Now, the number line is split into intervals you can test. When testing each interval, the objective is to substitute an x-value from the interval into the original inequality. This ensures that the solution set is accurately represented. Critical points are crucial because they enable us to focus on specific regions where the inequality might hold true.