Question

solve the following inequality. then graph the solution set.\\(|x| \leq 4\\\\\text{select the correct choices below, and, if necessary, fill in the answer box to complete your choice.}\\\\\text{a. the solution is an interval. the solution is } -4,4.\\\\\text{(simplify your answer. type your answer in interval notation. use integers or fractions for any numbers in the expression.)}\\\\\text{b. the solution set is one or two points. the solution set is } \\{\\ \\}.\\\\\text{(type an integer or a fraction. use a comma to separate answers as needed.)}\\\\\text{c. the solution set is } \varnothing.\\\\\text{choose the correct graph below}\\\\\text{a. }\cdots\text{ (graph with a dot at 4) }\cap\text{ b. }\cdots\text{ (graph with a line segment from -4 to 4)}

Explanation:

Step1: Recall the definition of absolute value inequality

The absolute value inequality \(|x| \leq a\) (where \(a \geq 0\)) is equivalent to \(-a \leq x \leq a\). For the inequality \(|x| \leq 4\), we can apply this definition.

Step2: Solve the compound inequality

From the definition, we get \(-4 \leq x \leq 4\). In interval notation, this is written as \([-4, 4]\). This means the solution is an interval (so option A is correct), and for the graph, we need to represent all real numbers from -4 to 4, including -4 and 4. So the graph should have closed circles at -4 and 4 and a line segment between them, which matches option B among the graph choices (assuming the second graph option B shows this).

Answer:

The solution to the inequality \(|x| \leq 4\) is the interval \([-4, 4]\) (so option A is correct: "A. The solution is an interval. The solution is \([-4, 4]\)"). For the graph, the correct choice is the one that represents the interval \([-4, 4]\) (with closed endpoints at -4 and 4 and shading between them, which is the second graph option, likely labeled as B in the given options).