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Use the drawing tools to form the correct answer on the number line.

Graph the solution set to this inequality:
[tex]\[ -2x + 9 \leq 5x - 12 \][/tex]


Sagot :

Sure! Let's solve the inequality step-by-step and then graph the solution on the number line.

1. Start with the given inequality:
[tex]\[ -2x + 9 \leq 5x - 12 \][/tex]

2. Isolate the variable [tex]\( x \)[/tex] on one side of the inequality. Let's move [tex]\( -2x \)[/tex] to the right-hand side by adding [tex]\( 2x \)[/tex] to both sides:
[tex]\[ 9 \leq 5x + 2x - 12 \][/tex]

3. Simplify the inequality:
[tex]\[ 9 \leq 7x - 12 \][/tex]

4. Next, add 12 to both sides to isolate terms involving [tex]\( x \)[/tex]:
[tex]\[ 9 + 12 \leq 7x \][/tex]

5. Simplify again:
[tex]\[ 21 \leq 7x \][/tex]

6. Finally, divide both sides by 7 to solve for [tex]\( x \)[/tex]:
[tex]\[ 3 \leq x \][/tex]

We can rewrite this as:
[tex]\[ x \geq 3 \][/tex]

Now, let's graph the solution [tex]\( x \geq 3 \)[/tex] on the number line.

1. Draw a number line.
2. Locate the critical point [tex]\( x = 3 \)[/tex] and mark it on the number line.
3. Since the inequality includes [tex]\( x = 3 \)[/tex], represent this with a closed (solid) dot at 3.
4. Shade the region to the right of 3 to indicate all values greater than or equal to 3.

Here is the graph on the number line:

[tex]\[ \begin{array}{ccccccccccccccccccc} & & & & & & & & & \bullet & \xrightarrow{\hspace{3cm}} & & & & & & & & \\ & - & 2 & & - & 1 & & 0 & & 1 & & 2 & & 3 & & 4 & & 5 & & \\ \end{array} \][/tex]

The solid dot at 3 and the shading to the right indicate that [tex]\( x \geq 3 \)[/tex] is the solution to the inequality.
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