Question

1. $y < |x - 5|$

$2x - y > 1$

1. $y < |x - 2|$

$y > 2x - 4$
9)

Explanation:

Problem 7: $y < |x-5|$ and $2x - y > 1$

Step1: Rewrite inequalities

$y < |x-5|$; $y < 2x - 1$

Step2: Graph $y=|x-5|$

This is a V-shaped graph with vertex at $(5,0)$, opening upwards. Draw it as a dashed line, shade below the line (since $y < |x-5|$).

Step3: Graph $y=2x-1$

This is a line with slope $2$, y-intercept $(0,-1)$. Draw it as a dashed line, shade below the line (since $y < 2x-1$).

Step4: Find intersection

Solve $|x-5|=2x-1$:

• Case 1: $x\geq5$: $x-5=2x-1 \implies x=-4$ (invalid, not $\geq5$)
• Case 2: $x<5$: $5-x=2x-1 \implies 3x=6 \implies x=2$, $y=2(2)-1=3$. Intersection at $(2,3)$.

Step5: Identify solution region

The overlapping shaded area is the solution: below both dashed lines, bounded by the V-shape and the straight line, with the intersection at $(2,3)$.

---

Problem 8: $y < |x-2|$ and $y > 2x - 4$

Step1: Rewrite inequalities

$y < |x-2|$; $y > 2x - 4$

Step2: Graph $y=|x-2|$

V-shaped graph with vertex at $(2,0)$, opening upwards. Draw as dashed line, shade below the line.

Step3: Graph $y=2x-4$

Line with slope $2$, y-intercept $(0,-4)$. Draw as dashed line, shade above the line (since $y > 2x-4$).

Step4: Find intersections

Solve $|x-2|=2x-4$:

• Case1: $x\geq2$: $x-2=2x-4 \implies x=2$, $y=0$. Intersection at $(2,0)$.
• Case2: $x<2$: $2-x=2x-4 \implies 3x=6 \implies x=2$ (invalid, not $<2$).

Find where $2x-4$ meets the left arm of the V: $2-x=2x-4$ gives the same $x=2$, so check another point: when $x=0$, $|0-2|=2$, $2(0)-4=-4$, so the region is between the line and the V-shape.

Answer:

For Problem 7:

• The solution is the overlapping shaded area below the dashed V-shaped graph $y=|x-5|$ and below the dashed line $y=2x-1$, with the boundary intersection at $(2,3)$.

For Problem 8:

• The solution is the overlapping shaded area below the dashed V-shaped graph $y=|x-2|$ and above the dashed line $y=2x-4$, with the boundary intersection at $(2,0)$.

(Note: To complete the task fully, plot the dashed lines and shade the overlapping regions on the provided coordinate grids as described.)