Decoding Compound Inequalities: Diagram and Solution for 40x+57≤5x-13≤25x+7

To solve the compound inequality $40x + 57 \leq 5x - 13 \leq 25x + 7$, we first break it down into two inequalities and solve them separately.

Step 1: Solve the inequality $40x + 57 \leq 5x - 13$.

Subtract $5x$ from both sides: $40x - 5x + 57 \leq -13$.

This simplifies to $35x + 57 \leq -13$.

Subtract 57 from both sides to isolate the term with $x$: $35x \leq -70$.

Divide by 35 to solve for $x$: $x \leq -2$.

Step 2: Solve the inequality $5x - 13 \leq 25x + 7$.

Subtract $5x$ from both sides: $-13 \leq 20x + 7$.

Subtract 7 from both sides to isolate the term with $x$: $-20 \leq 20x$.

Divide by 20 to solve for $x$: $-1 \leq x$.

Step 3: Determine the solution by finding the intersection of the two solutions.

The first solution is $x \leq -2$ and the second solution is $-1 \leq x$.

There is no overlap between $x \leq -2$ and $-1 \leq x$. Therefore, there is no value for $x$ that satisfies both conditions simultaneously.

Therefore, the solution is "No solution."

The correct diagram corresponds to choice 1, which indicates a solution of "No solution."