17. Douglas bought a \$20 game card at a game center. The go-karts cost \$3.50 each time you race. He wants to have at least $$ \$ 7.75 $$ left on his card to play arcade games. Write and solve an inequality to determine how many times Douglas can race the go-karts. Then interpret the solution. Define the variable: Inequality:

Question

UpStudy AI · Accepted Answer

Let's define the variable and set up the inequality step by step.

1. **Define the variable:**
   Let $$ x $$ be the number of times Douglas races the go-karts.

2. **Set up the inequality:**
   Douglas starts with \$20 and wants to have at least \$7.75 left after racing. The cost of each race is \$3.50. Therefore, the amount spent on racing is $$ 3.50x $$.

The inequality can be set up as follows:
   $$
   20 - 3.50x \geq 7.75
   $$

3. **Solve the inequality:**
   We will solve the inequality $$ 20 - 3.50x \geq 7.75 $$.

First, subtract 20 from both sides:
   $$
   -3.50x \geq 7.75 - 20
   $$

Now, calculate $$ 7.75 - 20 $$:
   $$
   -3.50x \geq -12.25
   $$

Next, divide both sides by -3.50. Remember that when dividing by a negative number, the inequality sign flips:
   $$
   x \leq \frac{-12.25}{-3.50}
   $$

Now, let's calculate $$ \frac{-12.25}{-3.50} $$:
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{-12.25}{\left(-3.5\right)}$$
- step1: Remove the parentheses:
  $$\frac{-12.25}{-3.5}$$
- step2: Convert the expressions:
  $$\frac{-\frac{49}{4}}{-\frac{7}{2}}$$
- step3: Multiply by the reciprocal:
  $$-\frac{49}{4}\left(-\frac{2}{7}\right)$$
- step4: Use the rules for multiplication and division:
  $$\frac{49}{4}\times \frac{2}{7}$$
- step5: Reduce the numbers:
  $$\frac{7}{2}\times 1$$
- step6: Multiply:
  $$\frac{7}{2}$$
The calculation gives us $$ \frac{7}{2} $$ or $$ 3.5 $$.

Thus, the inequality simplifies to:
$$
x \leq 3.5
$$

Since $$ x $$ represents the number of times Douglas can race, it must be a whole number. Therefore, the maximum number of times he can race is 3.

4. **Interpret the solution:**
Douglas can race the go-karts a maximum of 3 times while ensuring he has at least \$7.75 left on his game card for arcade games. If he races 4 times, he will not have enough money left for the arcade games.
Let $$ x $$ be the number of times Douglas races the go-karts. The inequality is: $$ 20 - 3.50x \geq 7.75 $$ Solving this, $$ x \leq 3.5 $$. Since $$ x $$ must be a whole number, Douglas can race the go-karts up to 3 times.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions