Question

1. (a)-(d), consider the inequality $x + 6 < -19$

a. solve the related equation to find the boundary number.
b. test a number for $x$ to determine which region to shade.
c. graph the solution set on the number line.
d. what is the solution set of the inequality?

1. for parts (a)-(d), consider the inequality $-13 \leq x - 5$

a. solve the related equation to find the boundary number.

Explanation:

Step1: Solve for boundary number

For $x + 6 = -19$:

$$\begin{align*} x &= -19 - 6 \\ x &= -25 \end{align*}$$

Step2: Test values for shading

Test $x=0$: $0+6=6 < -19$ is false.
Test $x=-30$: $-30+6=-24 < -19$ is true.

Step3: Graph solution set

Open circle at $x=-25$, shade left.

Step4: State solution set for first inequality

$x < -25$
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Step5: Solve boundary for second inequality

For $-13 = x - 5$:

$$\begin{align*} x &= -13 + 5 \\ x &= -8 \end{align*}$$

Step6: Test values for second inequality

Test $x=0$: $-13 \leq 0-5=-5$ is true.
Test $x=-10$: $-13 \leq -10-5=-15$ is false.

Step7: Graph second solution set

Closed circle at $x=-8$, shade right.

Step8: State solution set for second inequality

$x \geq -8$

Answer:

1. For inequality $x + 6 < -19$:

a. Boundary number: $x=-25$
b. Shade region left of $-25$ (test confirms validity)
c. Number line: Open circle at $-25$, arrow pointing left
d. Solution set: $x < -25$

1. For inequality $-13 \leq x - 5$:

a. Boundary number: $x=-8$
b. Shade region right of $-8$ (test confirms validity)
c. Number line: Closed circle at $-8$, arrow pointing right
d. Solution set: $x \geq -8$