Math  /  Algebra

QuestionSolve the inequality. Graph the solution set. 0.3x2.4>0.3|0.3 x-2.4|>0.3
Select the correct choice below and, if necessary, fill in the answer box to complete you A. The solution is one or more intervals. The solution is \square . (Type your answer in interval notation. Simplify your answer. Use integers or dec B. There are only one or two solutions. The solution set is \{ \square \}. (Type an integer or a fraction. Use a comma to separate answers as needed.) C. There is no solution.

Studdy Solution

STEP 1

1. The inequality involves an absolute value expression, which can be split into two separate inequalities.
2. The inequality 0.3x2.4>0.3|0.3x - 2.4| > 0.3 can be rewritten as two separate linear inequalities due to the properties of absolute values.
3. The solution set will be represented in interval notation.

STEP 2

1. Split the absolute value inequality into two separate inequalities.
2. Solve each of the two inequalities for x x .
3. Combine the solutions from the two inequalities into one solution set.
4. Graph the solution set on a number line.

STEP 3

Rewrite the absolute value inequality 0.3x2.4>0.3|0.3x - 2.4| > 0.3 as two separate inequalities:
0.3x2.4>0.30.3x - 2.4 > 0.3 0.3x2.4<0.30.3x - 2.4 < -0.3

STEP 4

Solve the inequality 0.3x2.4>0.30.3x - 2.4 > 0.3 for x x .
First, add 2.42.4 to both sides:
0.3x2.4+2.4>0.3+2.40.3x - 2.4 + 2.4 > 0.3 + 2.4
0.3x>2.70.3x > 2.7
Next, divide both sides by 0.30.3:
x>2.70.3x > \frac{2.7}{0.3}
x>9x > 9

STEP 5

Solve the inequality 0.3x2.4<0.30.3x - 2.4 < -0.3 for x x .
First, add 2.42.4 to both sides:
0.3x2.4+2.4<0.3+2.40.3x - 2.4 + 2.4 < -0.3 + 2.4
0.3x<2.10.3x < 2.1
Next, divide both sides by 0.30.3:
x<2.10.3x < \frac{2.1}{0.3}
x<7x < 7

STEP 6

Combine the solutions from the two inequalities.
The solution set is:
x<7orx>9x < 7 \quad \text{or} \quad x > 9
In interval notation, this is:
(,7)(9,)(-\infty, 7) \cup (9, \infty)

STEP 7

Graph the solution set on a number line.
- Draw a number line. - Shade the region to the left of x=7 x = 7 (excluding 7). - Shade the region to the right of x=9 x = 9 (excluding 9).
The graph will show that the solution set is:
(,7)(9,)(-\infty, 7) \cup (9, \infty)
Solution: The solution is one or more intervals. The solution is:
(,7)(9,)(-\infty, 7) \cup (9, \infty)

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