Solve the inequality by factoring. 3x2−7x+2<0 Note: Enter your answer and SHOW ALL THE STEPS NECESSARY to solve this problem in the space provided. You will not receive credit without all work shown. only solutions numbers..,

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To solve the inequality \( 3x^2 - 7x + 2 < 0 \) by factoring, we follow these steps:

Step 1: Factor the quadratic expression

We want to factor the quadratic \( 3x^2 - 7x + 2 \). We will look for two numbers that multiply to \( 3 \cdot 2 = 6 \) (the product of the coefficient of \( x^2 \) and the constant term) and add up to \( -7 \) (the coefficient of \( x \)).

The numbers that meet these criteria are \( -6 \) and \( -1 \) because:

  • \( -6 \cdot -1 = 6 \)
  • \( -6 + (-1) = -7 \)

Now we can rewrite the quadratic expression:

\[ 3x^2 - 6x - 1x + 2 \]

Next, we group the terms:

\[ (3x^2 - 6x) + (-1x + 2) \]

Factoring each group gives:

\[ 3x(x - 2) - 1(x - 2) \]

Now, we factor out \( (x - 2) \):

\[ (3x - 1)(x - 2) \]

Step 2: Set the factored expression to zero

To find the critical points, we set each factor equal to zero:

  1. \( 3x - 1 = 0 \) gives \( x = \frac{1}{3} \)
  2. \( x - 2 = 0 \) gives \( x = 2 \)

Step 3: Determine the intervals for testing

The critical points divide the number line into three intervals:

  1. \( (-\infty, \frac{1}{3}) \)
  2. \( \left(\frac{1}{3}, 2\right) \)
  3. \( (2, \infty) \)

Step 4: Test points from each interval

Let's test a point from each interval to determine where the expression \( (3x - 1)(x - 2) < 0 \).

  1. For \( x = 0 \) (in \( (-\infty, \frac{1}{3}) \)): \[ (3(0) - 1)(0 - 2) = (-1)(-2) = 2 > 0 \] The product is positive.

  2. For \( x = 1 \) (in \( \left(\frac{1}{3}, 2\right) \)): \[ (3(1) - 1)(1 - 2) = (3 - 1)(1 - 2) = (2)(-1) = -2 < 0 \] The product is negative.

  3. For \( x = 3 \) (in \( (2, \infty) \)): \[ (3(3) - 1)(3 - 2) = (9 - 1)(3 - 2) = (8)(1) = 8 > 0 \] The product is positive.

Step 5: Combine the results

The product \( (3x - 1)(x - 2) < 0 \) is true in the interval \( \left(\frac{1}{3}, 2\right) \).

Therefore, the solution to the inequality \( 3x^2 - 7x + 2 < 0 \) is:

\[ \boxed{\left(\frac{1}{3}, 2\right)} \]