Problem 59
Question
Solve each absolute value inequality. $$|x|<3$$
Step-by-Step Solution
Verified Answer
The solution to the inequality \(|x|<3\) is all x such that -3 < x < 3 or (-3,3) in interval notation.
1Step 1: Establish the Inequality Range
An absolute value of a number is always a non-negative number. It is the distance of the number from zero on the number line. Here, \(|x|<3\) means that x is within a distance of 3 from 0. It can be interpreted as: -3 < x < 3
2Step 2: Divide Into Two Inequalities
We can break the inequality -3 < x < 3, into two separate inequalities and solve them separately. These are: Inequality 1: -3 < x Inequality 2: x < 3
3Step 3: Plotting the Solution
On a number line, plot the solutions for both inequalities. The answer is the overlap or intersection of the two inequalities.
4Step 4: Write Final Answer
The final answer is the set of all x such that -3 < x < 3. In interval notation, this can be written as (-3,3).
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