Question

determine whether each of the integers from -5 to 5 is a solution of the inequality |x - 1| + 3 ≥ 5. if a number is a solution, plot it on the number line.

Explanation:

Step1: Simplify the inequality

Subtract 3 from both sides:
$|x-1| + 3 - 3 \geq 5 - 3$
$|x-1| \geq 2$

Step2: Break absolute value inequality

Split into two cases:
Case 1: $x-1 \geq 2$
Case 2: $x-1 \leq -2$

Step3: Solve Case 1

Isolate $x$:
$x \geq 2 + 1$
$x \geq 3$

Step4: Solve Case 2

Isolate $x$:
$x \leq -2 + 1$
$x \leq -1$

Step5: Test integers -5 to 5

Check which integers satisfy $x \leq -1$ or $x \geq 3$:

• $x=-5$: $|-5-1|+3=6+3=9\geq5$ ✔️
• $x=-4$: $|-4-1|+3=5+3=8\geq5$ ✔️
• $x=-3$: $|-3-1|+3=4+3=7\geq5$ ✔️
• $x=-2$: $|-2-1|+3=3+3=6\geq5$ ✔️
• $x=-1$: $|-1-1|+3=2+3=5\geq5$ ✔️
• $x=0$: $|0-1|+3=1+3=4<5$ ❌
• $x=1$: $|1-1|+3=0+3=3<5$ ❌
• $x=2$: $|2-1|+3=1+3=4<5$ ❌
• $x=3$: $|3-1|+3=2+3=5\geq5$ ✔️
• $x=4$: $|4-1|+3=3+3=6\geq5$ ✔️
• $x=5$: $|5-1|+3=4+3=7\geq5$ ✔️

Step6: Plot solutions on number line

Mark the integers $-5, -4, -3, -2, -1, 3, 4, 5$ on a number line with closed dots (since the inequality is $\geq$).

Answer:

The integers from -5 to 5 that are solutions are: $\boldsymbol{-5, -4, -3, -2, -1, 3, 4, 5}$

Number line representation:

<---•---•---•---•---•---o---o---o---•---•---•--->
   -5  -4  -3  -2  -1   0   1   2   3   4   5

(Closed dots • represent solutions; open dots o represent non-solutions)