To solve the inequality [tex]\( |-3x + 4| \leq 6 \)[/tex], we need to consider the definition of absolute value. The absolute value of a number is the distance from zero, which means it can be expressed in two parts as follows:
1. [tex]\(-3x + 4 \leq 6\)[/tex]
2. [tex]\(-3x + 4 \geq -6\)[/tex]
Let's solve each part separately.
Step 1: Solve the inequality [tex]\(-3x + 4 \leq 6\)[/tex]
1. Subtract 4 from both sides:
[tex]\[
-3x + 4 - 4 \leq 6 - 4
\][/tex]
This simplifies to:
[tex]\[
-3x \leq 2
\][/tex]
2. Divide both sides by -3. Remember, dividing by a negative number reverses the inequality:
[tex]\[
x \geq \frac{2}{-3}
\][/tex]
[tex]\[
x \geq -\frac{2}{3}
\][/tex]
So, one part of the solution is:
[tex]\[
x \geq -\frac{2}{3}
\][/tex]
Step 2: Solve the inequality [tex]\(-3x + 4 \geq -6\)[/tex]
1. Subtract 4 from both sides:
[tex]\[
-3x + 4 - 4 \geq -6 - 4
\][/tex]
This simplifies to:
[tex]\[
-3x \geq -10
\][/tex]
2. Divide both sides by -3. Remember, dividing by a negative number reverses the inequality:
[tex]\[
x \leq \frac{-10}{-3}
\][/tex]
[tex]\[
x \leq \frac{10}{3}
\][/tex]
So, the other part of the solution is:
[tex]\[
x \leq \frac{10}{3}
\][/tex]
Step 3: Combine the solutions
Combining both parts, we get:
[tex]\[
-\frac{2}{3} \leq x \leq \frac{10}{3}
\][/tex]
In interval notation, the solution to the inequality [tex]\( |-3x + 4| \leq 6 \)[/tex] is:
[tex]\[
\left[ -\frac{2}{3}, \frac{10}{3} \right]
\][/tex]
Thus, the values of [tex]\( x \)[/tex] that satisfy [tex]\( |-3x + 4| \leq 6 \)[/tex] lie within the interval:
[tex]\[
\left[ -\frac{2}{3}, \frac{10}{3} \right]
\][/tex]
