Question

solving linear equations

1. 3 - 4(3x + 2) = 31
2. 8x - 52 = 10 - 4x
3. x - 5 = -\frac{3}{2}x + \frac{5}{2}
4. \frac{4}{5} + \frac{2}{3}x = \frac{9}{10} - \frac{1}{6}x

solving linear inequalities

1. 3 - 2x > 11
2. 8(5 - 3x) < -24
3. how is x \leq 7 and x \geq 2 different from x \leq 7 or x \geq 2
4. solve 20 - 3x \geq 11 or -4x < -20 and explain what that means.

Explanation:

Question 22:

• For \( x \leq 7 \) and \( x \geq 2 \) (intersection of two sets):

The solution is the values of \( x \) that satisfy both inequalities simultaneously. This means \( x \) must be between 2 and 7 (inclusive), so \( 2 \leq x \leq 7 \). Geometrically, it’s the overlap of the two number lines (between 2 and 7).

• For \( x \leq 7 \) or \( x \geq 2 \) (union of two sets):

The solution is the values of \( x \) that satisfy at least one of the inequalities. Since every real number is either \( \leq 7 \), \( \geq 2 \), or both, this includes all real numbers (\( (-\infty, \infty) \)). Geometrically, it’s the combination of both number lines (no gaps).

Step 1: Solve \( 20 - 3x \geq 11 \)

Subtract 20 from both sides:
\( -3x \geq 11 - 20 \)
\( -3x \geq -9 \)

Divide both sides by \( -3 \) (reverse the inequality sign):
\( x \leq \frac{-9}{-3} \)
\( x \leq 3 \)

Step 2: Solve \( -4x < -20 \)

Divide both sides by \( -4 \) (reverse the inequality sign):
\( x > \frac{-20}{-4} \)
\( x > 5 \)

Step 3: Interpret “or”

The word “or” means the solution includes values that satisfy either \( x \leq 3 \) or \( x > 5 \) (no overlap, so combine both sets).

Answer:

• \( x \leq 7 \) and \( x \geq 2 \) means \( \boldsymbol{2 \leq x \leq 7} \) (only values between 2–7 inclusive).
• \( x \leq 7 \) or \( x \geq 2 \) means \( \boldsymbol{\text{all real numbers}} \) (every \( x \) satisfies at least one inequality).

Question 23: