Question

1. the cruz family shares a family cell phone plan. the plan is for 3,200 cell phone minutes each month. the father has used 1,200 minutes. the mother has used at least 600 minutes. the two children have used 675 minutes each. write an inequality that shows the number of minutes the cruz family has used. explain.

Explanation:

Step1: Define variables and identify usage

Let \( m \) be the minutes the mother used ( \( m \geq 600 \) ), father used 1200, each child used 675, so two children used \( 2\times675 \). Total used is father + mother + two children.

$$ 1200 + m + 2\times675 $$

Step2: Set up inequality based on plan limit

The total used must be less than or equal to 3200 (since the plan is 3200 minutes). Substitute the expressions:

$$ 1200 + m + 1350 \leq 3200 $$

Simplify the left - hand side:

$$ m + 2550 \leq 3200 $$

But since \( m\geq600 \), we can also write the inequality for total usage directly using the minimum for mother (600) to show the relationship. The total usage \( T = 1200 + m+675\times2 \), and \( T\leq3200 \), with \( m\geq600 \). Substituting the minimum \( m = 600 \) into the total usage formula for the inequality structure: \( 1200 + 600+675\times2\leq T\leq3200 \), but more accurately, since \( m\geq600 \), the inequality for total minutes used \( U \) is:

$$ 1200 + m+ 2\times675\leq3200 $$

Simplify \( 2\times675 = 1350 \), so \( 1200 + 1350+m\leq3200 \), \( 2550 + m\leq3200 \). But to write the inequality showing the number of minutes the family has used (in terms of the mother's minimum usage), we can use the fact that the mother used at least 600, so the total usage \( U=1200 + m + 675\times2\), and \( U\leq3200 \), and since \( m\geq600 \), we can also express the inequality as \( 1200 + 600+675\times2\leq U\leq3200 \) (where \( U \) is total usage). Calculating \( 1200 + 600+1350=3150 \), so \( 3150\leq U\leq3200 \) when using the minimum mother's usage, but the general inequality for \( U \) (total minutes used by the family) is:

$$ 1200 + m+ 2\times675\leq3200 $$

Or simplifying further:

$$ m + 2550\leq3200 $$

But if we want to write the inequality with the mother's usage as at least 600, we can also present it as:
Let \( U \) be the total minutes used by the family. Then \( U=1200 + m + 675\times2\), and since \( m\geq600 \), we have \( 1200 + 600+675\times2\leq U\leq3200 \), which simplifies to \( 3150\leq U\leq3200 \) (when considering the minimum mother's usage) or the more general inequality \( 1200 + m+1350\leq3200 \) with \( m\geq600 \).

Answer:

The inequality is \( 1200 + m+2\times675\leq3200 \) (or \( m + 2550\leq3200 \) with \( m\geq600 \)) where \( m \) is the minutes the mother used. In terms of total family usage \( U \), and using the mother's minimum usage, it can also be written as \( 1200 + 600+675\times2\leq U\leq3200 \) (simplifies to \( 3150\leq U\leq3200 \)) to show the relationship between the minimum total usage (with mother using at least 600) and the plan limit. The key idea is that the sum of the minutes used by father, mother, and two children must be less than or equal to the total minutes in the plan (3200), and since the mother used at least 600, we can construct the inequality based on that.