Task Sets: Represent constraints with equations - Common Core Math - HS Algebra

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Warm-Up: Review What You Know

These problems review skills from earlier lessons.

A car travels at 50 miles per hour. Which equation correctly relates distance $d$ (in miles) to time $t$ (in hours)?

$d + t = 50$

$d = 50t$

$t = 50d$

$50d + t = 0$

A point $(x, y)$ is a solution to $2x + 3y = 12$ . Which of the following points does NOT satisfy this equation?

$(0, 4)$

$(3, 2)$

$(6, 0)$

$(2, 3)$

Which of the following correctly describes the graph of $y \leq 2x + 1$ ?

A dashed line through $(0, 1)$ and $(1, 3)$ , with the region above the line shaded.

A solid line through $(0, 1)$ and $(1, 3)$ , with the region below the line shaded.

A dashed line through $(0, 1)$ and $(1, 3)$ , with the region below the line shaded.

A solid line through $(0, 1)$ and $(1, 3)$ , with the region above the line shaded.

Fluency Practice

A company makes product A and product B. Let $a$ = units of A produced and $b$ = units of B produced. Each unit of A requires 2 labor-hours; each unit of B requires 3 labor-hours. The company has at most 120 labor-hours available. Which inequality represents this constraint?

$2a + 3b \geq 120$

$2a + 3b \leq 120$

$2a + 3b = 120$

$3a + 2b \leq 120$

A nutritionist plans a meal using food X ( $10$ g protein/lb, $2/lb) and food Y ( $5$ g protein/lb, $4/lb). She needs at least 40 g protein and can spend at most $30. Let $x$ = pounds of X and $y$ = pounds of Y. Which system correctly models ALL the constraints, including non-negativity?

$10x + 5y \geq 40$ and $2x + 4y \leq 30$

$10x + 5y \geq 40$ , $2x + 4y \leq 30$ , $x \geq 0$ , $y \geq 0$

$10x + 5y \leq 40$ and $2x + 4y \leq 30$

$10x + 5y = 40$ and $2x + 4y = 30$

A bakery makes bread and muffins. Let $b$ = loaves of bread and $m$ = muffins. Each loaf uses 1 cup of flour; each muffin uses 0.5 cups. The bakery has 60 cups of flour. They must bake at least 40 items total. Which inequality represents the total items constraint?

$b \cdot m \geq 40$

$b + m \leq 40$

$b + m \geq 40$

$b - m \geq 40$

$Coordinate plane showing feasible region (light blue quadrilateral) bounded by x+y=6 and 2x+y=8, with four labeled candidate points A through D.$

The graph below shows the feasible region (shaded) for a system of two constraints plus non-negativity. The boundary lines are $x + y = 6$ (solid) and $2x + y = 8$ (solid). Which point is inside the feasible region?

$(5, 5)$

$(3, 4)$

$(2, 3)$

$(0, 7)$

A manufacturing constraint states that daily production of item P must be strictly more than 10 units. Which describes the boundary line for $p > 10$ on a graph where the horizontal axis represents $p$ ?

A solid vertical line at $p = 10$ , with the region to the right shaded.

A dashed vertical line at $p = 10$ , with the region to the left shaded.

A dashed vertical line at $p = 10$ , with the region to the right shaded.

A solid vertical line at $p = 10$ , with the region to the left shaded.

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Exercises: Represent Constraints with Equations, Inequalities, and Systems

Warm-Up: Review What You Know

Fluency Practice