Question

prompt: solve the compound inequality x - 2 < -5 or 2x + 3 > 9. make a claim about how the word “or” affects the solution set. use evidence from this example by writing the solution in interval notation and showing it on a number line. then explain your reasoning by describing why the solution includes values from both sides rather than overlapping values.

Explanation:

Step1: Solve the first inequality

Add 2 to both sides of $x - 2 < - 5$.
$x-2 + 2< - 5+2$
$x < - 3$

Step2: Solve the second inequality

Subtract 3 from both sides of $2x + 3>9$:
$2x+3 - 3>9 - 3$
$2x>6$
Divide both sides by 2:
$x > 3$

Step3: Analyze the 'or' condition

The word 'or' means that a value is in the solution - set if it satisfies either the first inequality or the second inequality.

Step4: Write the solution in interval notation

The solution in interval notation is $(-\infty,-3)\cup(3,\infty)$.

Step5: Reasoning about the solution

The solution includes values from both sides rather than overlapping values because the 'or' connective means that we take all the values that make either of the inequalities true. There is no overlap between $x < - 3$ and $x > 3$, so we combine the non - overlapping intervals.

Answer:

The solution of the compound inequality is $x < - 3$ or $x > 3$, in interval notation $(-\infty,-3)\cup(3,\infty)$. On a number line, we have an open circle at $x=-3$ and shade to the left, and an open circle at $x = 3$ and shade to the right. The 'or' word means we consider values that satisfy either part of the compound inequality, so we combine the two non - overlapping regions.