Make a sign chart that represents the expression left parenthesis, 2, x, plus, 4, right parenthesis, left parenthesis, minus, 8, x, plus, 7, right parenthesis, is less than or equal to, 0(2x+4)(−8x+7)≤0 and use your sign chart to solve the inequality.

To solve the inequality \( (2x + 4)(-8x + 7) \leq 0 \), we will use a **sign chart** to determine where the expression is negative or zero. Here's a step-by-step breakdown of how to approach this:

### Step 1: Find the critical points
First, we need to solve for the values of \(x\) that make each factor equal to zero. These are the critical points where the sign of the expression can change.

1. \( 2x + 4 = 0 \)

   \[
   2x = -4 \quad \Rightarrow \quad x = -2
   \]

2. \( -8x + 7 = 0 \)

   \[
   -8x = -7 \quad \Rightarrow \quad x = \frac{7}{8}
   \]

So, the critical points are \( x = -2 \) and \( x = \frac{7}{8} \).

### Step 2: Make a sign chart
We'll divide the real number line into intervals based on these critical points and test each interval to determine the sign of the expression \( (2x + 4)(-8x + 7) \).

The intervals are:
- \( (-\infty, -2) \)
- \( (-2, \frac{7}{8}) \)
- \( (\frac{7}{8}, \infty) \)

### Step 3: Test the sign in each interval

1. **Interval \( (-\infty, -2) \)**: Choose \( x = -3 \)

   \[
   2(-3) + 4 = -6 + 4 = -2 \quad \text{(negative)}
   \]
   \[
   -8(-3) + 7 = 24 + 7 = 31 \quad \text{(positive)}
   \]
   \[
   \text{Product: } (-2)(31) = -62 \quad \text{(negative)}
   \]

2. **Interval \( (-2, \frac{7}{8}) \)**: Choose \( x = 0 \)

   \[
   2(0) + 4 = 4 \quad \text{(positive)}
   \]
   \[
   -8(0) + 7 = 7 \quad \text{(positive)}
   \]
   \[
   \text{Product: } (4)(7) = 28 \quad \text{(positive)}
   \]

3. **Interval \( (\frac{7}{8}, \infty) \)**: Choose \( x = 1 \)

   \[
   2(1) + 4 = 6 \quad \text{(positive)}
   \]
   \[
   -8(1) + 7 = -8 + 7 = -1 \quad \text{(negative)}
   \]
   \[
   \text{Product: } (6)(-1) = -6 \quad \text{(negative)}
   \]

### Step 4: Check the values at the critical points

- At \( x = -2 \):

   \[
   (2(-2) + 4)(-8(-2) + 7) = (0)(23) = 0
   \]

- At \( x = \frac{7}{8} \):

   \[
   (2(\frac{7}{8}) + 4)(-8(\frac{7}{8}) + 7) = (2)(0) = 0
   \]

Both critical points make the expression equal to 0.

### Step 5: Interpret the sign chart and solve the inequality

We want the product to be **less than or equal to zero**. This includes both the negative intervals and the points where the product is zero.

- The expression is **negative** on \( (-\infty, -2) \) and \( (\frac{7}{8}, \infty) \).
- The expression is **zero** at \( x = -2 \) and \( x = \frac{7}{8} \).

### Solution:
The solution to \( (2x + 4)(-8x + 7) \leq 0 \) is:

\[
x \in [-2, \frac{7}{8}]
\]

This includes the closed interval because the product equals zero at the endpoints.

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Do you want further details on how the intervals were chosen, or do you have any questions?

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### Related Questions:

1. How do you solve quadratic inequalities graphically?
2. How do you determine where a rational function is positive or negative?
3. What is the significance of critical points in solving inequalities?
4. How does the concept of intervals help in solving inequalities?
5. How would the sign chart change if the inequality were strict (less than 0)?

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### Tip:
When using sign charts, always test values within intervals to determine the sign of the product or quotient—this will help accurately identify solution regions.

Make a sign chart that represents the expression (2x + 4)(-8x + 7) ≤ 0 and use your sign chart to solve the inequality.

Learn how to solve the inequality (2x + 4)(-8x + 7) ≤ 0 by creating and interpreting a sign chart. This step-by-step guide covers critical points, interval testing, and the solution, suitable for students in Grades 9-11. Understand key algebraic concepts such as the zero product property and the sign chart method.

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