Question

1. solve and graph the inequality (-\frac{1}{3}x < -\frac{3}{4}x - 5 + \frac{5}{2}x)

Explanation:

Step1: Combine like terms on the right

First, combine the \( x \)-terms on the right side: \( -\frac{3}{4}x + \frac{5}{2}x \). To add these, get a common denominator (4): \( -\frac{3}{4}x + \frac{10}{4}x = \frac{7}{4}x \). So the inequality becomes \( -\frac{1}{3}x < \frac{7}{4}x - 5 \).

Step2: Subtract \( \frac{7}{4}x \) from both sides

Subtract \( \frac{7}{4}x \) from each side to get all \( x \)-terms on the left: \( -\frac{1}{3}x - \frac{7}{4}x < -5 \). Find a common denominator (12) for the \( x \)-terms: \( -\frac{4}{12}x - \frac{21}{12}x = -\frac{25}{12}x \). So now we have \( -\frac{25}{12}x < -5 \).

Step3: Multiply both sides by \( -\frac{12}{25} \) (reverse inequality)

Multiply both sides by \( -\frac{12}{25} \). Remember, when multiplying or dividing an inequality by a negative number, reverse the inequality sign: \( x > (-5) \times (-\frac{12}{25}) \). Simplify the right side: \( (-5) \times (-\frac{12}{25}) = \frac{60}{25} = \frac{12}{5} = 2.4 \).

Answer:

\( x > \frac{12}{5} \) (or \( x > 2.4 \))
To graph: Draw an open circle at \( \frac{12}{5} \) (or 2.4) on the number line and shade to the right.