Question

solve the linear inequality. other than \\( \emptyset \\), use interval notation to express the solution set and graph the solution set on a number line.\\( 5(x + 5) \geq 4(x - 4) + x \\)\\(\leftarrow\frac{}{-10 9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10}\
ightarrow\\)\\( \bigcirc \\ (-\infty, 4 \\)\\(\leftarrow\frac{}{-10 9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10}\
ightarrow\\)\\( \bigcirc \\ 4, \infty) \\)\\(\leftarrow\frac{}{-10 9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10}\
ightarrow\\)\\( \bigcirc \\ (-\infty, \infty) \\)\\(\leftarrow\frac{}{-10 9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10}\
ightarrow\\)\\( \bigcirc \\ \emptyset \\)\\(\leftarrow\frac{}{-10 9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10}\
ightarrow\\)

Explanation:

Step1: Expand both sides

$5x + 25 \geq 4x - 16 + x$

Step2: Simplify right-hand side

$5x + 25 \geq 5x - 16$

Step3: Subtract $5x$ from both sides

$25 \geq -16$

Step4: Analyze the resulting statement

This is always true for all real $x$.

Answer:

$\boldsymbol{(-\infty, \infty)}$
(The corresponding graph is the third option, where the entire number line is shaded with arrows pointing in both directions.)