Solve the Inequality for x |3-2x|> =4
The question is asking you to find all the values of the variable x that satisfy the inequality involving the absolute value expression |3 - 2x| being greater than or equal to 4. The solution will involve considering two cases based on the definition of absolute value: one where the quantity inside the absolute value (3 - 2x) is non-negative, and another where it is negative, and then solving the resulting inequalities to find the range of values for x that meet the given condition.
$\left|\right. 3 - 2 x \left|\right. \geq 4$
Answer
Solution:
Step 1
Express the inequality $\left| 3 - 2x \right| \geq 4$ using a piecewise approach.
Step 1.1
Identify the range where the expression inside the absolute value is non-negative: $3 - 2x \geq 0$.
Step 1.2
Resolve the inequality.
Step 1.2.1
Subtract $3$ from both sides: $-2x \geq -3$.
Step 1.2.2
Divide by $-2$ and remember to reverse the inequality: $x \leq \frac{3}{2}$.
Step 1.3
For the case where $3 - 2x \geq 0$, the absolute value can be removed: $3 - 2x \geq 4$.
Step 1.4
Determine the interval where the expression inside the absolute value is negative: $3 - 2x < 0$.
Step 1.5
Address the inequality.
Step 1.5.1
Subtract $3$ from both sides: $-2x < -3$.
Step 1.5.2
Divide by $-2$ and flip the inequality: $x > \frac{3}{2}$.
Step 1.6
For the case where $3 - 2x < 0$, remove the absolute value and multiply by $-1$: $-(3 - 2x) \geq 4$.
Step 1.7
Represent as a piecewise function.
Step 1.8
Simplify the expression $-(3 - 2x) \geq 4$.
Step 2
Solve the inequality $3 - 2x \geq 4$ for $x$.
Step 2.1
Isolate terms involving $x$.
Step 2.1.1
Subtract $3$ from both sides: $-2x \geq 1$.
Step 2.1.2
Combine constants: $-2x \geq 1$.
Step 2.2
Divide by $-2$ and reverse the inequality: $x \leq -\frac{1}{2}$.
Step 3
Resolve $-3 + 2x \geq 4$ for $x$.
Step 3.1
Shift terms not containing $x$.
Step 3.1.1
Add $3$ to both sides: $2x \geq 7$.
Step 3.1.2
Combine constants: $2x \geq 7$.
Step 3.2
Divide by $2$: $x \geq \frac{7}{2}$.
Step 4
Combine the solutions: $x \leq -\frac{1}{2}$ or $x \geq \frac{7}{2}$.
Step 5
Present the solution in various formats.
Inequality Form: $x \leq -\frac{1}{2}$ or $x \geq \frac{7}{2}$ Interval Notation: $(-\infty, -\frac{1}{2}] \cup [\frac{7}{2}, \infty)$
Knowledge Notes:
To solve an absolute value inequality like $\left| 3 - 2x \right| \geq 4$, we must consider two cases because the absolute value function outputs the distance from zero, which can be achieved by either a positive or negative quantity inside the absolute value.
Piecewise Approach: This involves breaking down the inequality into two separate cases based on the sign of the expression inside the absolute value.
Inequality Manipulation: When solving inequalities, if we multiply or divide by a negative number, we must reverse the direction of the inequality sign.
Simplification: After isolating the variable, we simplify the expressions to find the solution set.
Combining Solutions: Since the absolute value inequality can have two cases, we find the union of the solution sets from both cases.
Interval Notation: This is a way of describing the set of solutions in terms of intervals. For example, $(-\infty, -\frac{1}{2}]$ means all numbers up to and including $-\frac{1}{2}$.
Absolute Value Properties: The absolute value of a number is always non-negative, and $\left| x \right| \geq a$ implies that $x \leq -a$ or $x \geq a$ for some positive $a$.
By considering these points, we can effectively solve absolute value inequalities and represent the solution in various forms.
