Question

do you remember? solving and graphing a one - variable inequality
a) solve the inequality - 2x + 3 ≤ 9 for x.
b) graph the solution set on a number line.
how do you determine which direction the graph goes?
how do you know what type of circle to place on the number line?

Explanation:

Step1: Isolate the variable term

Subtract 3 from both sides of -2x + 3 ≤ 9.
-2x+3 - 3≤9 - 3, which simplifies to -2x≤6.

Step2: Solve for x

Divide both sides of -2x≤6 by -2. When dividing an inequality by a negative number, the direction of the inequality sign changes. So, $\frac{-2x}{-2}\geq\frac{6}{-2}$, resulting in x≥ - 3.

Step3: Graphing on number - line

For x≥ - 3, we place a closed circle at - 3 (because the inequality is "greater than or equal to", so - 3 is included in the solution set) and draw an arrow to the right (since x is greater than or equal to - 3).

The direction of the graph is determined by the sign of the inequality. If it is "greater than" (>) or "greater than or equal to" (≥), the graph goes to the right. If it is "less than" (<) or "less than or equal to" (≤), the graph goes to the left.

The type of circle is determined by the inequality symbol. A closed circle (●) is used for "less than or equal to" (≤) and "greater than or equal to" (≥) because the value at that point is included in the solution set. An open circle (○) is used for "less than" (<) and "greater than" (>) because the value at that point is not included in the solution set.

Answer:

a) x≥ - 3
b) Place a closed circle at - 3 and draw an arrow to the right on the number - line.