Questions: Solve the inequality. Graph the solution set, simplified fractions or integers. (5+w)/12-(w+2)/8>-1/48

$Solve the inequality. Graph the solution set, simplified fractions or integers. (5+w)/12-(w+2)/8>-1/48$

Transcript text: Solve the inequality. Graph the solution set, simplified fractions or integers. \[ \frac{5+w}{12}-\frac{w+2}{8}>-\frac{1}{48} \] Part: $0 / 3$

Solution

Solution Steps

Step 1: Simplify the Inequality

First, we need to find a common denominator for the fractions. The denominators are 12, 8, and 48. The least common multiple of these numbers is 48.

\[ \frac{5+w}{12} = \frac{4(5+w)}{48} = \frac{20+4w}{48} \]

\[ \frac{w+2}{8} = \frac{6(w+2)}{48} = \frac{6w+12}{48} \]

Substitute these into the inequality:

\[ \frac{20+4w}{48} - \frac{6w+12}{48} > -\frac{1}{48} \]

Step 2: Combine the Fractions

Combine the fractions on the left side:

\[ \frac{20+4w - (6w+12)}{48} > -\frac{1}{48} \]

Simplify the numerator:

\[ \frac{20 + 4w - 6w - 12}{48} > -\frac{1}{48} \]

\[ \frac{8 - 2w}{48} > -\frac{1}{48} \]

Step 3: Clear the Denominator

Multiply both sides by 48 to clear the denominator:

\[ 8 - 2w > -1 \]

Step 4: Solve for $ w $

Add 1 to both sides:

\[ 8 - 2w + 1 > 0 \]

\[ 9 - 2w > 0 \]

Subtract 9 from both sides:

\[ -2w > -9 \]

Divide by -2 and reverse the inequality sign:

\[ w < \frac{9}{2} \]

Final Answer

The solution to the inequality is:

\[ w < \frac{9}{2} \]

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = (9/2)"], "latex_expressions": ["$w < \\frac{9}{2}$"]}

Was this solution helpful?