Question

you try: solve and graph the inequality below. make sure you show all your work!

1. $7 < -5 + x$

is 12 a solution?
2.
is 27 a solution?

1. $-2 \leq x - 3$

is 1 a solution?

1. $-6 \leq \frac{1}{2}x - 2$

is $-4$ a solution?

1. $8 + x < -6$

is $-13$ a solution?

1. $x - -4 \leq -13$

is $-17$ a solution?

Explanation:

Step1: Solve inequality 1

Isolate $x$ by adding 5 to both sides.
$7 + 5 < -5 + x + 5$
$12 < x$ or $x > 12$
Check if 12 is a solution: Substitute $x=12$ into $7 < -5 + x$: $7 < -5+12 \implies 7<7$, which is false.

Step2: Solve inequality 2

Multiply both sides by 3 to isolate $x$.
$\frac{x}{3} \times 3 \geq 9 \times 3$
$x \geq 27$
Check if 27 is a solution: Substitute $x=27$ into $\frac{x}{3} \geq 9$: $\frac{27}{3} \geq 9 \implies 9\geq9$, which is true.

Step3: Solve inequality 3

Isolate $x$ by adding 3 to both sides.
$-2 + 3 \leq x - 3 + 3$
$1 \leq x$ or $x \geq 1$
Check if 1 is a solution: Substitute $x=1$ into $-2 \leq x - 3$: $-2 \leq 1-3 \implies -2\leq-2$, which is true.

Step4: Solve inequality 4

Isolate $x$ by subtracting 6 from both sides.
$-6 - 6 \leq \frac{1}{2}x + 6 - 6$
$-12 \leq \frac{1}{2}x$
Multiply both sides by 2: $-12 \times 2 \leq \frac{1}{2}x \times 2$
$x \geq -24$
Check if $-4$ is a solution: Substitute $x=-4$ into $-6 \leq \frac{1}{2}x + 6$: $-6 \leq \frac{-4}{2} + 6 \implies -6 \leq -2 + 6 \implies -6\leq4$, which is true.

Step5: Solve inequality 5

Isolate $x$ by subtracting 8 from both sides.
$8 + x - 8 < -6 - 8$
$x < -14$
Check if $-13$ is a solution: Substitute $x=-13$ into $8 + x < -6$: $8 + (-13) < -6 \implies -5 < -6$, which is false.

Step6: Solve inequality 6

Simplify and isolate $x$: $x + 4 \leq -13$
Subtract 4 from both sides: $x + 4 - 4 \leq -13 - 4$
$x \leq -17$
Check if $-17$ is a solution: Substitute $x=-17$ into $x - (-4) \leq -13$: $-17 + 4 \leq -13 \implies -13\leq-13$, which is true.

Answer:

1. Solution: $\boldsymbol{x > 12}$; Is 12 a solution? $\boldsymbol{\text{No}}$
2. Solution: $\boldsymbol{x \geq 27}$; Is 27 a solution? $\boldsymbol{\text{Yes}}$
3. Solution: $\boldsymbol{x \geq 1}$; Is 1 a solution? $\boldsymbol{\text{Yes}}$
4. Solution: $\boldsymbol{x \geq -24}$; Is -4 a solution? $\boldsymbol{\text{Yes}}$
5. Solution: $\boldsymbol{x < -14}$; Is -13 a solution? $\boldsymbol{\text{No}}$
6. Solution: $\boldsymbol{x \leq -17}$; Is -17 a solution? $\boldsymbol{\text{Yes}}$

Graph Notes (for reference):

1. Open circle at 12, arrow pointing right
2. Closed circle at 27, arrow pointing right
3. Closed circle at 1, arrow pointing right
4. Closed circle at -24, arrow pointing right
5. Open circle at -14, arrow pointing left
6. Closed circle at -17, arrow pointing left