Which graph represents the solution to $5+8 x<3(2 x+4)$?
We are presented with the linear inequality \(5 + 8x < 3(2x + 4)\). Our goal is to determine the range of values for \(x\) that satisfy this condition. To do this, we need to manipulate the expression algebraically to isolate \(x\) on one side of the inequality. We will follow standard algebraic principles to simplify the inequality step by step.
First, we distribute the constant 3 on the right-hand side of the inequality:
\[3(2x + 4) = 6x + 12\]
Substituting this back into the inequality, we get:
\[5 + 8x < 6x + 12\]
Next, we gather all the terms containing \(x\) on one side by subtracting \(6x\) from both sides:
\[5 + 8x - 6x < 12\]
\[5 + 2x < 12\]
Now, we isolate the \(2x\) term by subtracting 5 from both sides:
\[2x < 12 - 5\]
\[2x < 7\]
Finally, we solve for \(x\) by dividing both sides by 2:
\[x < 3.5\]
The solution to the inequality is \(x < 3.5\). In interval notation, this is represented as \((-\infty, 3.5)\). This means that any value less than 3.5 is a solution to the inequality. To represent this graphically on a number line, we place an open circle at 3.5 (since 3.5 is not included in the solution set, as indicated by the strict inequality <) and shade the number line to the left, towards negative infinity.
Imagine you are setting up a budget for a monthly subscription service. You have an initial setup fee of \$5 and an additional cost of \$8 per user. You compare this to another plan that costs \$3 per user plus a fixed monthly overhead fee of \$4. If you represent the number of users by \(x\), the inequality \(5 + 8x < 3(2x + 4)\) helps you determine the range of users for which the first plan is cheaper than the second. Finding this threshold allows you to make informed financial decisions based on your expected number of users.