Question

solve for x and graph the solution on the number line below: ( 20 < 5x + 0 ) and ( 60 > 5x + 0 )
answer: attempt 1 out of 5
inequality notation: ( 4 < x < 12 )
number line: number line with -12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, 12, ... (click and drag to plot lines)
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Explanation:

Step1: Solve the first inequality \(20 < 5x + 6\)

Subtract 6 from both sides: \(20 - 6 < 5x + 6 - 6\)
Simplify: \(14 < 5x\)
Divide both sides by 5: \(\frac{14}{5} < x\)? Wait, no, wait, let's check again. Wait, the correct first step: \(20 < 5x + 6\), subtract 6: \(20 - 6 < 5x\) → \(14 < 5x\)? Wait, no, the given inequality notation is \(4 < x < 12\), so let's do it correctly. Wait, \(20 < 5x + 6\): subtract 6 from both sides: \(20 - 6 < 5x\) → \(14 < 5x\)? No, that's not right. Wait, maybe I miscalculated. Wait, \(5x + 6 > 20\) (same as \(20 < 5x + 6\)): subtract 6: \(5x > 20 - 6\) → \(5x > 14\)? No, the answer is \(4 < x\), so maybe the first inequality is \(20 < 5x + 6\), let's solve it:

\(20 < 5x + 6\)

Subtract 6 from both sides: \(20 - 6 < 5x\) → \(14 < 5x\)? No, that would be \(x > 14/5 = 2.8\), but the answer is \(x > 4\). Wait, maybe the first inequality is \(20 < 5x + 6\) is wrong? Wait, no, the second inequality is \(66 > 5x + 6\), let's solve that:

\(66 > 5x + 6\)

Subtract 6: \(60 > 5x\)

Divide by 5: \(12 > x\) → \(x < 12\)

Now the first inequality: let's assume the first inequality is \(20 < 5x + 6\), but that gives \(x > 14/5 = 2.8\), but the answer is \(x > 4\). Wait, maybe the first inequality is \(26 < 5x + 6\)? Wait, the image shows "26 < 5x + 6 and 66 > 5x + 6". Oh! I misread the first inequality. It's \(26 < 5x + 6\), not 20. Let's correct that.

Step1: Solve \(26 < 5x + 6\)

Subtract 6 from both sides: \(26 - 6 < 5x\)
Simplify: \(20 < 5x\)
Divide both sides by 5: \(\frac{20}{5} < x\) → \(4 < x\)

Step2: Solve \(66 > 5x + 6\)

Subtract 6 from both sides: \(66 - 6 > 5x\)
Simplify: \(60 > 5x\)
Divide both sides by 5: \(\frac{60}{5} > x\) → \(12 > x\) or \(x < 12\)

Step3: Combine the two inequalities

From Step1, \(x > 4\); from Step2, \(x < 12\). So the solution is \(4 < x < 12\)

For the number line: We need to plot an open circle at 4 (since \(x > 4\), not \(x \geq 4\)) and an open circle at 12 (since \(x < 12\), not \(x \leq 12\)), then draw a line segment between them to represent all numbers between 4 and 12.

Answer:

Inequality Notation: \(4 < x < 12\)

Number Line: Open circles at 4 and 12, with a line connecting them (representing all real numbers between 4 and 12).