Problem 11 - Calculus: One and Several Variables

Question

Solve the inequality and mark the solution set on a number line. $$2 x^{2}+x-1 \leq 0$$.

Step-by-Step Solution

Answer

The solution set is \( x \in [-\infty, 0.5] \)

Step 1: Factoring the quadratic expression

Unfortunately, the quadratic expression \(2x^2 + x - 1\) cannot be factored using integers. Therefore, we use the quadratic formula \(x = [-b ± sqrt(b^2 - 4ac)] / 2a \) to find the roots of the equation.

Step 2: Calculate the roots

Using the quadratic formula, the roots of the equation are calculated as: \(x = [-1 ± sqrt((1)^2 - 4*2*(-1))] / 2*2 = [-1 ± sqrt(1 + 8)] / 4 = [-1 ± sqrt(9)] / 4 = [-1 ± 3 ] / 4 = 2/4, -4/4\) which simplifies to \( x = 0.5, -1 \) . These are the roots of the equation \(2x^2 + x - 1 = 0\) and these split the the number line into three intervals.

Step 3: Identify the intervals to test

The roots split the number line into three intervals: \(-\infty < x < -1\), \(-1 < x < 0.5\), \(0.5 < x < \infty\). For each of these intervals, we pick a test point to check if the inequality \(2x^2 + x - 1 \leq 0\) holds.

Step 4: Test the intervals

We take test points, say -2, 0, and 1 for each of the intervals respectively. For x = -2, we substitute into the inequality to get \(2*(-2)^2 - 2 - 1 \leq 0\) which simplifies to \(-3 \leq 0\) . This is true so the first interval \(-\infty < x < -1\) is part of the solution. For x = 0, we substitute into the inequality to get \( -1 \leq 0 \). This is true so the second interval \(-1 < x < 0.5\) is also part of the solution. For x = 1, we substitute into the inequality to get \(2*1^2 + 1 - 1 \leq 0\) which simplifies to \( 2 \leq 0 \). This is false so the third interval \(0.5 < x < \infty\) is not part of the solution.

Step 5: Mark the intervals on the number line

The solution to the inequality is the union of the two intervals and the roots i.e., the final answer is \( x \in [-\infty, -1] \cup [-1, 0.5] = [-\infty, 0.5] \) . This set is marked on a number line.

Key Concepts

Understanding the Quadratic Formula

The quadratic formula is a reliable method to find the roots of any quadratic equation. A quadratic equation is generally in the form of \( ax^2 + bx + c = 0 \). Here, the coefficients \( a \), \( b \), and \( c \) represent real numbers, and \( a \) should not be equal to zero as it would otherwise not be a quadratic equation anymore.
The formula itself is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The expression under the square root, \( b^2 - 4ac \), is known as the discriminant. It helps identify the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If zero, there is one real, repeated root.
• If negative, the equation has two complex (non-real) roots.

For our inequality \( 2x^2 + x - 1 \leq 0 \), using the quadratic formula finds the roots as \( x = 0.5 \) and \( x = -1 \). These roots help determine the intervals for testing the inequality.

Finding Inequality Solutions

Solving inequalities is slightly different from solving equations. Instead of finding exact solutions, we look for ranges (or sets of solutions) that satisfy the inequality. The inequality given, \(2x^2 + x - 1 \leq 0\), is already set up to solve. However, dealing with inequalities requires that we consider the sign changes within those ranges determined by our found roots.
Here's how it works:

• Identify the roots (as we did using the quadratic formula) which split the number line into intervals. These were \(-1\) and \(0.5\).
• In each interval, choose a test value and substitute it back into the inequality.
• If the inequality holds true for that test value, then the whole interval is part of the solution set.

Remember:

• Use an appropriate test point that lies within each interval.
• Solutions to inequalities involve inequalities themselves, meanings other than "equals".
• If the inequality is \(\leq\) or \(\geq\), the boundary points (roots) are included in the solution set.

Here, \(-1 \leq x \leq 0.5\) satisfied the inequality, including both roots as part of the solution.

Interpreting the Number Line Representation

Once we have determined which intervals satisfy the inequality, plotting these on a number line gives a visual representation of the solution set. This step helps in verifying our solution easily, especially in exams or practical applications.
Here's how to mark solutions on a number line:

• Draw a horizontal line and put small marks (points) at the roots you found, \(-1\) and \(0.5\) in this case.
• Shade or draw a solid line between the intervals that satisfy the inequality, including the boundary points if applicable.
• Each segment of the line within the solution set and boundary points signifies the range of \(x\) values that make the inequality true.

For our inequality, this means shading from \(-\infty\) to \(0.5\). Such shading indicates precisely where \( x \) falls in making the inequality \(2x^2 + x - 1 \leq 0\) true.
Visualizing inequalities on a number line is excellent for quickly confirming that solutions are understood and correct.