Questions: Solve the following inequality and graph the solution set. -4x + 10 <= 9

Transcript text: Quiz Solve the following inequality and graph the solution set. \[ |-4 x+10| \leq 9 \]

Solution

Solution Steps

Step 1: Solve the absolute value inequality

To solve the inequality $|-4x + 10| \leq 9$, we need to consider two cases:

$-4x + 10 \leq 9$
$-(-4x + 10) \leq 9$ which simplifies to $4x - 10 \leq 9$

Step 2: Solve the first case

\[ -4x + 10 \leq 9 \] Subtract 10 from both sides: \[ -4x \leq -1 \] Divide by -4 (and reverse the inequality sign): \[ x \geq \frac{1}{4} \]

Step 3: Solve the second case

\[ 4x - 10 \leq 9 \] Add 10 to both sides: \[ 4x \leq 19 \] Divide by 4: \[ x \leq \frac{19}{4} \]

Final Answer

Combining both cases, the solution set is: \[ \frac{1}{4} \leq x \leq \frac{19}{4} \]

{"axisType": 3, "coordSystem": {"xmin": -1, "xmax": 6, "ymin": -1, "ymax": 2}, "commands": ["y = 0", "y = 1"], "latex_expressions": ["$\\frac{1}{4} \\leq x \\leq \\frac{19}{4}$"]}

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