Question

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solve the inequality. express your answer using set notation and interval notation. graph the solution set.
-4(x + 3) < 16

the solution is expressed in set notation as {x | □}.
the solution is expressed in interval notation as □.
choose the correct graph of the solution set below.
a. graph with a bracket at -7 and arrow to the right
b. graph with a parenthesis at -7 and arrow to the right
c. graph with an arrow to the left ending at -7 with a parenthesis
d. graph with an arrow to the left ending at -7 with a bracket

Explanation:

Step1: Divide both sides by -4 (reverse inequality)

Divide each side of \(-4(x + 3) < 16\) by \(-4\). Remember, when dividing by a negative number, the inequality sign flips. So we get \(x + 3 > \frac{16}{-4}\), which simplifies to \(x + 3 > -4\).

Step2: Subtract 3 from both sides

Subtract 3 from both sides of the inequality \(x + 3 > -4\). This gives \(x + 3 - 3 > -4 - 3\), so \(x > -7\).

Set Notation:

The solution in set notation is \(\{x | x > -7\}\) because we're describing all real numbers \(x\) such that \(x\) is greater than -7.

Interval Notation:

For interval notation, since \(x\) is greater than -7 (not including -7), we use \((-7, \infty)\). The parenthesis around -7 indicates it's not included, and \(\infty\) (infinity) is always paired with a parenthesis.

Graph Analysis:

We need a number line with an open circle at -7 (since -7 is not included in the solution) and an arrow pointing to the right (because \(x\) is greater than -7). Looking at the options:

• Option A: Closed circle at -7 (incorrect, as -7 is not included).
• Option B: Open circle at -7, arrow to the right (correct).
• Option C: Arrow to the left (incorrect, since \(x > -7\) is to the right of -7).
• Option D: Closed circle at -7, arrow to the left (incorrect).

Answer:

Set notation: \(\{x | x > -7\}\)
Interval notation: \((-7, \infty)\)
Graph: Option B (Open circle at -7, arrow to the right)