Question

level 3
graph the compound inequality.

1. $x\geq - 3$ and $x < 5$
2. $x\leq - 4$ or $x\geq 2$
3. the amount of snowfall in a night was at least 3 inches but less than 7 inches. graph the possible values of snowfall on the number line.

inches of snowfall
solve each equation. must show all steps for credit.

1. $7x + 8-2x = 63$
2. $4x + 1 = 2x+31$
3. $2(3x - 34)-4 = 2x - 40$

solve and graph each inequality. must show all steps and work for credit.

1. $5x + 2\geq - 23$
2. $-\frac{x}{4}+4 > 8$

Explanation:

11)

Step1: Analyze inequality \(x\geq - 3\)

On the number - line, mark a closed circle at \(-3\) (because \(x\) can equal \(-3\)) and shade to the right.

Step2: Analyze inequality \(x < 5\)

On the same number - line, mark an open circle at \(5\) (because \(x\) cannot equal \(5\)) and shade to the left. The overlapping shaded region is the solution.

12)

Step1: Analyze inequality \(x\leq - 4\)

On the number - line, mark a closed circle at \(-4\) and shade to the left.

Step2: Analyze inequality \(x\geq 2\)

On the same number - line, mark a closed circle at \(2\) and shade to the right. The union of the two shaded regions is the solution.

13)

Step1: Translate the statement

The statement "at least 3 inches but less than 7 inches" can be written as \(3\leq x<7\).

Step2: Graph on number - line

Mark a closed circle at \(3\) and an open circle at \(7\), then shade the region between them.

14)

Step1: Combine like terms

\(7x + 8-2x=63\) simplifies to \(5x+8 = 63\) (since \(7x-2x = 5x\)).

Step2: Subtract 8 from both sides

\(5x+8 - 8=63 - 8\), so \(5x=55\).

Step3: Divide both sides by 5

\(\frac{5x}{5}=\frac{55}{5}\), then \(x = 11\).

15)

Step1: Subtract \(2x\) from both sides

\(4x + 1-2x=2x + 31-2x\), which gives \(2x+1 = 31\).

Step2: Subtract 1 from both sides

\(2x+1 - 1=31 - 1\), so \(2x=30\).

Step3: Divide both sides by 2

\(\frac{2x}{2}=\frac{30}{2}\), then \(x = 15\).

16)

Step1: Distribute the 2

\(2(3x - 34)-4=2x-40\) becomes \(6x-68 - 4=2x-40\), which simplifies to \(6x-72=2x-40\).

Step2: Subtract \(2x\) from both sides

\(6x-2x-72=2x-2x - 40\), so \(4x-72=-40\).

Step3: Add 72 to both sides

\(4x-72 + 72=-40 + 72\), then \(4x = 32\).

Step4: Divide both sides by 4

\(\frac{4x}{4}=\frac{32}{4}\), so \(x = 8\).

17)

Step1: Subtract 2 from both sides

\(5x+2-2\geq - 23-2\), so \(5x\geq - 25\).

Step2: Divide both sides by 5

\(\frac{5x}{5}\geq\frac{-25}{5}\), then \(x\geq - 5\). On the number - line, mark a closed circle at \(-5\) and shade to the right.

18)

Step1: Subtract 4 from both sides

\(-\frac{x}{4}+4 - 4>8 - 4\), so \(-\frac{x}{4}>4\).

Step2: Multiply both sides by - 4 and reverse the inequality sign

\(-\frac{x}{4}\times(-4)<4\times(-4)\), then \(x<-16\). On the number - line, mark an open circle at \(-16\) and shade to the left.

Answer:

1. The solution on the number - line has a closed circle at \(-3\), an open circle at \(5\), and the region between them is shaded.
2. The solution on the number - line has a closed circle at \(-4\) with shading to the left and a closed circle at \(2\) with shading to the right.
3. The solution on the number - line has a closed circle at \(3\), an open circle at \(7\), and the region between them is shaded.
4. \(x = 11\)
5. \(x = 15\)
6. \(x = 8\)
7. \(x\geq - 5\), with a closed - circle at \(-5\) and shading to the right on the number - line.
8. \(x<-16\), with an open - circle at \(-16\) and shading to the left on the number - line.