Question

question 1 of 34 step 2 of 2
consider the following compound inequality.
\\(\frac{20}{4} < \frac{x + 4}{2} < \frac{25}{4}\\)
step 2 of 2: graph the solution set.
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pressing the buttons will create an interval with the indicated endpoint type. the interval is graphed by moving the endpoints to the desired values. to remove graphed intervals, use the clear button. to indicate infinity or negative infinity, drag an open end point to the end of the number line.
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Explanation:

Step1: Solve the left inequality

First, simplify \(\frac{20}{4}\) to \(5\), so the left part is \(5 < \frac{x + 4}{2}\). Multiply both sides by \(2\): \(5\times2 < x + 4\), which gives \(10 < x + 4\). Then subtract \(4\) from both sides: \(10 - 4 < x\), so \(6 < x\).

Step2: Solve the right inequality

Simplify \(\frac{25}{4}\) to \(6.25\), so the right part is \(\frac{x + 4}{2} < 6.25\). Multiply both sides by \(2\): \(x + 4 < 6.25\times2\), which is \(x + 4 < 12.5\). Subtract \(4\) from both sides: \(x < 12.5 - 4\), so \(x < 8.5\).

Step3: Determine the interval and graph

The solution to the compound inequality is \(6 < x < 8.5\), which is the interval \((6, 8.5)\). On the number line, we use an open circle at \(6\) (since \(x > 6\), not \(x\geq6\)) and an open circle at \(8.5\) (since \(x < 8.5\), not \(x\leq8.5\)), then draw a line between them.

Answer:

The solution set is the interval \((6, 8.5)\), graphed with open circles at \(6\) and \(8.5\) and a line connecting them on the number line.