Question

solve the linear inequality graphically. write the solution set in set-builder notation.$-2(x - 1290) + 52 geq 61$graph the inequality. all graphs are $-2000, 2000, 100$ by $-50, 100, 10$. choose the correct graph below.$\bigcirc$ a.$\bigcirc$ b.$\bigcirc$ c.$\bigcirc$ d.

Explanation:

Step1: Expand the left-hand side

$-2(x - 1290) + 52 = -2x + 2580 + 52$

Step2: Simplify the constant terms

$-2x + 2580 + 52 = -2x + 2632$

Step3: Isolate the variable term

$-2x + 2632 \geq 61 \\
-2x \geq 61 - 2632$

Step4: Calculate the constant difference

$-2x \geq -2571$

Step5: Solve for x (reverse inequality)

$x \leq \frac{-2571}{-2} \\
x \leq 1285.5$

Step6: Identify the correct graph

The solution is a vertical dashed line at $x=1285.5$ with shading to the left (since $x$ is less than or equal to this value). This matches option C.

Answer:

Set-builder notation: $\{x \mid x \leq 1285.5\}$
Correct graph: C