Question

9-33. solve each equation. be sure to find all possible solutions and check your solutions. homework help a. |x| = 7 b. |2x| = 32 c. |x + 7| = 10 d. 5|x| = 35 9-34. determine if the following statements are true or false. homework help a. |-6| < 4 b. |-3 + 5| > 2.5 c. 4 ≥ |0| d. |-4 + 3| > 1

Explanation:

9-33. Solve each equation

a. \(|x| = 7\)

Step1: Recall absolute value definition

The absolute value of a number \(x\), \(|x|\), is the distance from \(x\) to \(0\) on the number line. So if \(|x| = 7\), then \(x\) can be \(7\) or \(-7\) because both \(7\) and \(-7\) are \(7\) units away from \(0\).

Step2: Write the solutions

So the solutions are \(x = 7\) or \(x = -7\).

Step1: Apply absolute value property

If \(|A| = B\) (where \(B\geq0\)), then \(A = B\) or \(A = -B\). Here \(A = 2x\) and \(B = 32\), so \(2x = 32\) or \(2x = -32\).

Step2: Solve for \(x\) in each equation

For \(2x = 32\), divide both sides by \(2\): \(x=\frac{32}{2}=16\). For \(2x = -32\), divide both sides by \(2\): \(x=\frac{-32}{2}=-16\).

Step1: Use absolute value property

If \(|A| = B\) (\(B\geq0\)), then \(A = B\) or \(A = -B\). Let \(A=x + 7\) and \(B = 10\), so \(x + 7=10\) or \(x + 7=-10\).

Step2: Solve each linear equation

For \(x + 7=10\), subtract \(7\) from both sides: \(x=10 - 7 = 3\). For \(x + 7=-10\), subtract \(7\) from both sides: \(x=-10 - 7=-17\).

Answer:

\(x = 7\) or \(x = -7\)

b. \(|2x| = 32\)