Symbolize and Solve Linear Inequalities

Question

Jordan wants to spend no more than $30 at the bowling alley. The bowling alley charges a flat fee of $10 to enter. They also charge $4 per game. Write an inequality for the number of games (x) that Jordan can play and describe the solutions.

Answer 1

Total spent on activity ≤ Total cash available to spend

Given a $4 fee per game.

g = the number of games bowled.

The total spent for bowling g games is 4(g), (This does not include the $10 entrance fee).

Given that Jordan does not want to spend more than $30 at the bowling alley, the equation will include $30 on the right side of the equation with a less than or equal to inequality sign positioned immediately before the $30 value.

Total spent for bowling g games + Entrance Fee ≤ Total Available to Spend

4g + 10 ≤ 30

SOLUTION:

4g + 10 ≤ 30

4g ≤ 20 (Subtract 10 from each side)

g ≤ 5 (Divide each side by 4)

The number of games bowled is 5 or any number of games less than 5.

Noteworthy (not part of the solution):

In this example, we did not divide both sides of the equation by a negative integer; instead we divided both sides by positive 4. Notice the inequality sign did not change. (We did not 'flip it'.) If it had been necessary to divide both sides of the equation by a negative integer to arrive at the solution, then we would have had to change the inequality sign FROM: 'less than or equal to' TO: 'greater than or equal to'. (FROM: ≤ TO: ≥ )

Answer 2

You can describe this situation in the form f(x) = m(x) + b, where x is the number of games played, m is the price per game, b is the flat fee, and f(x) is the total money spent. This is because each game played (x) will increase the money spent by m, and b is a charge that's independent of the number of games. Plug in the values for the price limit, the price per game, and the flat fee, and you'll get the inequality ($4)(x) + $10 ≤ $30. Subtract $10 from each side to get ($4)(x) ≤ $20, and then divide both sides by $4 to get x ≤ 5. Jordan can't play fewer than zero games, so the minimum possible value for x is 0. The maximum is the solution we found which is 5, where the money spent would be f(5) = ($4)(5) + $10 = $20 + $10 = $30, the maximum acceptable amount. Our answer for number of games that can be played is thus 0 ≤ x ≤ 5.

Answer 3

4x+10 ≤ 30

4x ≤ 20

x ≤ 5

So he can play five games or less and stay within his budget.