Question

for the system below, determine the solution set that satisfies the system, find the intercepts, and determine if it is solid or dashed line.
solution:
4x - y ≥ 0
x + y ≤ 5
x - intercept
(10,0)
(0,0)
(-5,0)
(0,8)
(0,0)
(5,0)
y - intercept
(0,0)
(0,0)
(0,5)
line type
solid
solid
shade
below
below

Explanation:

Step1: Analyze \(4x - y \geq 0\)

Rewrite as \(y \leq 4x\). For intercepts, set \(x = 0\), \(y = 0\); set \(y = 0\), \(x = 0\). The inequality has \(\geq\), so solid line. Shade below \(y = 4x\).

Step2: Analyze \(x + y \leq 5\)

Rewrite as \(y \leq -x + 5\). X - intercept: set \(y = 0\), \(x = 5\) (so \((5,0)\)). Y - intercept: set \(x = 0\), \(y = 5\) (so \((0,5)\)). Inequality has \(\leq\), so solid line. Shade below \(y=-x + 5\).

Step3: Solution Set

Find intersection of regions. Test \((0,0)\): satisfies both \(4(0)-0\geq0\) and \(0 + 0\leq5\). So solution includes \((0,0)\) and region where both inequalities hold.

Answer:

• Solution: Region satisfying \(y \leq 4x\) and \(y \leq -x + 5\) (includes \((0,0)\) etc.).
• \(x\)-intercepts: For \(4x - y \geq 0\) is \((0,0)\); for \(x + y \leq 5\) is \((5,0)\).
• \(y\)-intercepts: For \(4x - y \geq 0\) is \((0,0)\); for \(x + y \leq 5\) is \((0,5)\).
• Line types: Both solid (due to \(\geq\) and \(\leq\)).
• Shade: Below both lines.