To solve the inequality involving an absolute value, we need to consider the definition of the absolute value function. Specifically, for a given expression [tex]\( |A| \leq B \)[/tex], this implies:
[tex]\[ -B \leq A \leq B \][/tex]
In our case, the inequality is given as:
[tex]\[ \left| \frac{4}{7} + 2x \right| \leq \frac{2}{5} \][/tex]
We can set up the inequality as two separate inequalities without the absolute value:
[tex]\[ -\frac{2}{5} \leq \frac{4}{7} + 2x \leq \frac{2}{5} \][/tex]
We'll solve these two inequalities step by step.
### Solve the first inequality:
[tex]\[ -\frac{2}{5} \leq \frac{4}{7} + 2x \][/tex]
Subtract [tex]\(\frac{4}{7}\)[/tex] from both sides of the inequality:
[tex]\[ -\frac{2}{5} - \frac{4}{7} \leq 2x \][/tex]
To combine the fractions, we need a common denominator. The common denominator for 5 and 7 is 35.
Convert [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{4}{7}\)[/tex] to have the common denominator:
[tex]\[ -\frac{2}{5} = -\frac{2 \cdot 7}{5 \cdot 7} = -\frac{14}{35} \][/tex]
[tex]\[ \frac{4}{7} = \frac{4 \cdot 5}{7 \cdot 5} = \frac{20}{35} \][/tex]
Now, combine the fractions:
[tex]\[ -\frac{14}{35} - \frac{20}{35} \leq 2x \][/tex]
[tex]\[ -\frac{34}{35} \leq 2x \][/tex]
To isolate [tex]\(x\)[/tex], divide both sides by 2:
[tex]\[ -\frac{34}{35 \cdot 2} \leq x \][/tex]
[tex]\[ -\frac{34}{70} \leq x \][/tex]
[tex]\[ -\frac{17}{35} \leq x \][/tex]
### Solve the second inequality:
[tex]\[ \frac{4}{7} + 2x \leq \frac{2}{5} \][/tex]
Subtract [tex]\(\frac{4}{7}\)[/tex] from both sides:
[tex]\[ 2x \leq \frac{2}{5} - \frac{4}{7} \][/tex]
Again, convert to a common denominator of 35:
[tex]\[ \frac{2}{5} = \frac{2 \cdot 7}{5 \cdot 7} = \frac{14}{35} \][/tex]
[tex]\[ \frac{4}{7} = \frac{4 \cdot 5}{7 \cdot 5} = \frac{20}{35} \][/tex]
Now, subtract the fractions:
[tex]\[ 2x \leq \frac{14}{35} - \frac{20}{35} \][/tex]
[tex]\[ 2x \leq -\frac{6}{35} \][/tex]
Divide both sides by 2:
[tex]\[ x \leq -\frac{6}{35 \cdot 2} \][/tex]
[tex]\[ x \leq -\frac{6}{70} \][/tex]
[tex]\[ x \leq -\frac{3}{35} \][/tex]
### Combine the solutions:
Now, we combine the results from both inequalities. From the first inequality, we have:
[tex]\[ -\frac{17}{35} \leq x \][/tex]
From the second inequality, we have:
[tex]\[ x \leq -\frac{3}{35} \][/tex]
Together, we get:
[tex]\[ -\frac{17}{35} \leq x \leq -\frac{3}{35} \][/tex]
So, the solution to the inequality [tex]\( \left| \frac{4}{7} + 2x \right| \leq \frac{2}{5} \)[/tex] is:
[tex]\[ -\frac{17}{35} \leq x \leq -\frac{3}{35} \][/tex]
In the form requested:
[tex]\[ -\frac{17}{35} \leq x \leq -\frac{3}{35} \][/tex]
Therefore:
[tex]\[ -\frac{17}{35} \leq x \leq -\frac{3}{35} \][/tex]
