Problem 7 - Algebra for College Students

Question

Solve each inequality and graph its solution set on a number line. $$(x+2)(4 x-3) \leq 0$$

Step-by-Step Solution

Answer

The solution is \([-2, \frac{3}{4}]\).

Step 1: Identify Critical Points

To solve the inequality \((x+2)(4x-3) \leq 0\), first find the critical points by setting each factor equal to zero.For \(x+2 = 0\), solve to get \(x = -2\).For \(4x-3 = 0\), solve to get \(x = \frac{3}{4}\).

Step 2: Determine Intervals

The critical points divide the number line into intervals: \((-\infty, -2)\), \((-2, \frac{3}{4})\), and \((\frac{3}{4}, \infty)\). These intervals will be used to test the sign of the inequality.

Step 3: Test Each Interval

Choose a test point in each interval to determine the sign of \((x+2)(4x-3)\).1. For \((-\infty, -2)\), choose \(x = -3\): \((x+2)(4x-3) = (-3+2)(4(-3)-3) = (-1)(-15) = 15\).2. For \((-2, \frac{3}{4})\), choose \(x = 0\): \((x+2)(4x-3) = 2(-3) = -6\).3. For \((\frac{3}{4}, \infty)\), choose \(x = 1\): \((x+2)(4x-3) = 3(1) = 3\).

Step 4: Determine the Solution Set

Since the inequality is \((x+2)(4x-3) \leq 0\), we are looking for intervals where the expression is \(\leq 0\). These are the intervals where the product is negative or zero.From Step 3, \((-2, \frac{3}{4})\) is the interval where the product is negative. Also, because the inequality is \(\leq 0\), including the critical points where the product is zero, the solution set is \([-2, \frac{3}{4}]\).

Step 5: Graph the Solution on a Number Line

Draw a number line and mark the critical points \(-2\) and \(\frac{3}{4}\) with closed circles (since these points satisfy the inequality \(\leq 0\)). Then shade the interval between these points to indicate the solutions include all values from \(-2\) to \(\frac{3}{4}\), inclusive.

Key Concepts

Critical Points

When solving inequalities like \((x+2)(4x-3) \leq 0\), critical points are essential for determining where the function changes sign. Critical points arise by setting each factor of the expression \((x+2)\) and \((4x-3)\) equal to zero. This helps us find values of \(x\) where the expression could change from positive to negative or vice versa.

• For \(x+2=0\), solving gives \(x=-2\).
• For \(4x-3=0\), solving gives \(x=\frac{3}{4}\).

These values divide the number line into distinct intervals for further testing. Remember, critical points are not just turning points in inequalities; they can also be points where the expression is zero.

Interval Testing

Once critical points are identified, the next step is interval testing. This technique involves selecting test points from the different intervals created by the critical points and evaluating the sign of the inequality.
To carry out interval testing for \((x+2)(4x-3)\), divide the number line into the following intervals: \((-\infty, -2)\), \((-2, \frac{3}{4})\), and \((\frac{3}{4}, \infty)\).

• In \((-\infty, -2)\), use \(x = -3\); this evaluates to a positive product of \(15\).
• In \((-2, \frac{3}{4})\), use \(x = 0\); this evaluates to a negative product of \(-6\).
• In \((\frac{3}{4}, \infty)\), use \(x = 1\); this evaluates to a positive product of \(3\).

By testing these intervals, we can discern which portions of the number line solve the inequality.

Solution Set

To determine the solution set of an inequality, examine the results of interval testing to see where the inequality holds true. We are interested in where \((x+2)(4x-3) \leq 0\), which means the expression can be negative or zero.
From the interval testing, only \((-2, \frac{3}{4})\) yields a negative result. Since the inequality includes \(\leq 0\), the critical points \(-2\) and \(\frac{3}{4}\) should also be included because they make the expression equal zero.
Thus, the solution set becomes \([-2, \frac{3}{4}]\). Understanding that the critical points satisfied the equality portion of the inequality is crucial as it reflects on solving real-world problems involving constraints.

Number Line Graphing

Graphing the solution set on a number line provides a visual representation of intervals where the inequality is satisfied. This makes it easier to convey where the solutions lie.
Follow these steps to graph the solution:Mark the critical points \(-2\) and \(\frac{3}{4}\) with closed circles. These points are included in the solution set because the inequality allows for equality (\(\leq 0\)). Next, shade the interval between \(-2\) and \(\frac{3}{4}\), indicating all numbers within this segment satisfy the inequality.
In summary:

• Closed circles at \(-2\) and \(\frac{3}{4}\).
• Shaded portion of number line between \(-2\) and \(\frac{3}{4}\).

These steps ensure the graphical approach complements the algebraic solution by clearly illustrating the values satisfying the inequality.