Answer:
Explanation:
To solve the absolute value inequality |2x + 4| ≤ 6, we need to find all possible values of x that satisfy the inequality. By the definition of absolute value, we know that the expression |2x + 4| will be less than or equal to 6 if we square it. Therefore, we can solve:
(2x + 4)^2 ≤ 36.
Simplifying this expression, we get:
4x^2 + 16x + 16 ≤ 36.
4x^2 + 12x − 20 ≤ 0.
Discriminant: D = 12^2 - 4(4)(−20) < 0, so there are no real solutions.
Therefore, the solution is the empty set ∅.
The interval notation is an alternative to the set notation, which represents intervals or ranges of numbers. The interval notation for the solution set of the inequality is [-∞, ∞). This means that all real numbers, both positive and negative, satisfy the inequality.
The graph of the absolute value inequality is the set of points (x, y) such that |2x + 4| ≤ 6. This graph is simply a straight line with slope 2 and passing through the point (-2, 0).