Modeling with Equation/InequalityConstraints - Algebra
Card 1 of 30
Find the correct constraint for “$x$ is no more than $5$ more than $y$.”
Find the correct constraint for “$x$ is no more than $5$ more than $y$.”
Tap to reveal answer
$x \le y + 5$. Upper limit on how much one exceeds another.
$x \le y + 5$. Upper limit on how much one exceeds another.
← Didn't Know|Knew It →
What inequality represents “$x$ is nonnegative” in a modeling context?
What inequality represents “$x$ is nonnegative” in a modeling context?
Tap to reveal answer
$x \ge 0$. Cannot be negative in most contexts.
$x \ge 0$. Cannot be negative in most contexts.
← Didn't Know|Knew It →
What does the constraint “between $3$ and $7$, not inclusive” translate to?
What does the constraint “between $3$ and $7$, not inclusive” translate to?
Tap to reveal answer
$3 < x < 7$. Range excludes both endpoint values.
$3 < x < 7$. Range excludes both endpoint values.
← Didn't Know|Knew It →
What does the constraint “between $3$ and $7$, inclusive” translate to?
What does the constraint “between $3$ and $7$, inclusive” translate to?
Tap to reveal answer
$3 \le x \le 7$. Range includes both endpoint values.
$3 \le x \le 7$. Range includes both endpoint values.
← Didn't Know|Knew It →
What does the constraint “exactly $50$” translate to as an equation?
What does the constraint “exactly $50$” translate to as an equation?
Tap to reveal answer
$= 50$. Precise value with no variation allowed.
$= 50$. Precise value with no variation allowed.
← Didn't Know|Knew It →
What does the constraint “no less than $50$” translate to using an inequality?
What does the constraint “no less than $50$” translate to using an inequality?
Tap to reveal answer
$\ge 50$. Minimum threshold that must be met.
$\ge 50$. Minimum threshold that must be met.
← Didn't Know|Knew It →
Which line style is used for $<$ or $>$ when graphing an inequality?
Which line style is used for $<$ or $>$ when graphing an inequality?
Tap to reveal answer
A dashed boundary line. Points on the line don't satisfy the constraint.
A dashed boundary line. Points on the line don't satisfy the constraint.
← Didn't Know|Knew It →
Identify the constraint for “$x$ exceeds $y$ by at least $3$.”
Identify the constraint for “$x$ exceeds $y$ by at least $3$.”
Tap to reveal answer
$x - y \ge 3$. Difference must meet the minimum threshold.
$x - y \ge 3$. Difference must meet the minimum threshold.
← Didn't Know|Knew It →
Identify the constraint for “$x$ is within $3$ of $y$” using an absolute value inequality.
Identify the constraint for “$x$ is within $3$ of $y$” using an absolute value inequality.
Tap to reveal answer
$|x-y| \le 3$. Distance between values stays within bounds.
$|x-y| \le 3$. Distance between values stays within bounds.
← Didn't Know|Knew It →
What two inequalities are equivalent to the constraint $|x-y| \le 3$?
What two inequalities are equivalent to the constraint $|x-y| \le 3$?
Tap to reveal answer
$-3 \le x-y \le 3$. Absolute value inequalities split into compound form.
$-3 \le x-y \le 3$. Absolute value inequalities split into compound form.
← Didn't Know|Knew It →
What inequality represents “profit is at least $\$0$” if profit is $P$?
What inequality represents “profit is at least $\$0$” if profit is $P$?
Tap to reveal answer
$P \ge 0$. Break-even or better financial performance.
$P \ge 0$. Break-even or better financial performance.
← Didn't Know|Knew It →
Identify the constraint for “$x$ is at most $5$ less than $y$.”
Identify the constraint for “$x$ is at most $5$ less than $y$.”
Tap to reveal answer
$x \ge y - 5$. Maximum amount one can fall below another.
$x \ge y - 5$. Maximum amount one can fall below another.
← Didn't Know|Knew It →
Identify the constraint for “revenue minus cost is nonnegative” if revenue is $R$ and cost is $C$.
Identify the constraint for “revenue minus cost is nonnegative” if revenue is $R$ and cost is $C$.
Tap to reveal answer
$R - C \ge 0$. Profitable operation requirement.
$R - C \ge 0$. Profitable operation requirement.
← Didn't Know|Knew It →
Which phrase translates to the inequality $x \ge 12$ in context?
Which phrase translates to the inequality $x \ge 12$ in context?
Tap to reveal answer
“At least $12$.”. Minimum value requirement phrase.
“At least $12$.”. Minimum value requirement phrase.
← Didn't Know|Knew It →
Which phrase translates to the inequality $x \le 12$ in context?
Which phrase translates to the inequality $x \le 12$ in context?
Tap to reveal answer
“At most $12$.”. Maximum value limitation phrase.
“At most $12$.”. Maximum value limitation phrase.
← Didn't Know|Knew It →
Identify whether the point $(2,3)$ is viable for the system $x\ge 0$, $y\ge 0$, $x+y\le 4$.
Identify whether the point $(2,3)$ is viable for the system $x\ge 0$, $y\ge 0$, $x+y\le 4$.
Tap to reveal answer
Nonviable, because $2+3>4$. Sum exceeds the constraint limit.
Nonviable, because $2+3>4$. Sum exceeds the constraint limit.
← Didn't Know|Knew It →
Identify whether the point $(1,3)$ is viable for the system $x\ge 0$, $y\ge 0$, $x+y\le 4$.
Identify whether the point $(1,3)$ is viable for the system $x\ge 0$, $y\ge 0$, $x+y\le 4$.
Tap to reveal answer
Viable, because $1+3=4$ and both are nonnegative. Satisfies all system requirements exactly.
Viable, because $1+3=4$ and both are nonnegative. Satisfies all system requirements exactly.
← Didn't Know|Knew It →
Identify whether $(4,0)$ satisfies the constraint $2x + y \le 7$.
Identify whether $(4,0)$ satisfies the constraint $2x + y \le 7$.
Tap to reveal answer
Nonviable, because $2(4)+0=8>7$. Substitution shows constraint violation.
Nonviable, because $2(4)+0=8>7$. Substitution shows constraint violation.
← Didn't Know|Knew It →
Identify whether $(3,1)$ satisfies the constraint $2x + y \le 7$.
Identify whether $(3,1)$ satisfies the constraint $2x + y \le 7$.
Tap to reveal answer
Viable, because $2(3)+1=7$. Satisfies the constraint exactly at boundary.
Viable, because $2(3)+1=7$. Satisfies the constraint exactly at boundary.
← Didn't Know|Knew It →
What is the feasible region for a system of constraints?
What is the feasible region for a system of constraints?
Tap to reveal answer
The set of all points satisfying every constraint. All valid solutions within constraint boundaries.
The set of all points satisfying every constraint. All valid solutions within constraint boundaries.
← Didn't Know|Knew It →
In a system, what does it mean if a point satisfies all but one inequality?
In a system, what does it mean if a point satisfies all but one inequality?
Tap to reveal answer
It is nonviable because one violated constraint is enough. Any constraint violation makes solution invalid.
It is nonviable because one violated constraint is enough. Any constraint violation makes solution invalid.
← Didn't Know|Knew It →
What is the standard first step to test whether $(a,b)$ satisfies an inequality constraint?
What is the standard first step to test whether $(a,b)$ satisfies an inequality constraint?
Tap to reveal answer
Substitute $x=a$ and $y=b$ and check if it is true. Direct substitution verifies constraint satisfaction.
Substitute $x=a$ and $y=b$ and check if it is true. Direct substitution verifies constraint satisfaction.
← Didn't Know|Knew It →
What does the inequality symbol $\le$ mean in a constraint?
What does the inequality symbol $\le$ mean in a constraint?
Tap to reveal answer
$\le$ means “less than or equal to.”. Allows equality at the boundary value.
$\le$ means “less than or equal to.”. Allows equality at the boundary value.
← Didn't Know|Knew It →
What does the inequality symbol $\ge$ mean in a constraint?
What does the inequality symbol $\ge$ mean in a constraint?
Tap to reveal answer
$\ge$ means “greater than or equal to.”. Allows equality at the boundary value.
$\ge$ means “greater than or equal to.”. Allows equality at the boundary value.
← Didn't Know|Knew It →
Identify the constraint for “$x$ is at least twice $y$.”
Identify the constraint for “$x$ is at least twice $y$.”
Tap to reveal answer
$x \ge 2y$. Sets lower bound as double the other variable.
$x \ge 2y$. Sets lower bound as double the other variable.
← Didn't Know|Knew It →
Identify the constraint for “$x$ is no more than twice $y$.”
Identify the constraint for “$x$ is no more than twice $y$.”
Tap to reveal answer
$x \le 2y$. Sets upper bound as double the other variable.
$x \le 2y$. Sets upper bound as double the other variable.
← Didn't Know|Knew It →
What inequality models “use at most $60$ minutes” if task 1 takes $5x$ minutes and task 2 takes $8y$ minutes?
What inequality models “use at most $60$ minutes” if task 1 takes $5x$ minutes and task 2 takes $8y$ minutes?
Tap to reveal answer
$5x + 8y \le 60$. Total time cannot exceed the available time.
$5x + 8y \le 60$. Total time cannot exceed the available time.
← Didn't Know|Knew It →
What inequality models “at least $8$ hours total” if activity A is $x$ hours and B is $y$ hours?
What inequality models “at least $8$ hours total” if activity A is $x$ hours and B is $y$ hours?
Tap to reveal answer
$x + y \ge 8$. Combined time must meet the minimum.
$x + y \ge 8$. Combined time must meet the minimum.
← Didn't Know|Knew It →
What inequality models “no more than $500$ calories” if foods give $120x$ and $200y$ calories?
What inequality models “no more than $500$ calories” if foods give $120x$ and $200y$ calories?
Tap to reveal answer
$120x + 200y \le 500$. Total calorie intake has an upper bound.
$120x + 200y \le 500$. Total calorie intake has an upper bound.
← Didn't Know|Knew It →
What inequality models “at least $40$ grams of protein” if foods give $6x$ and $9y$ grams?
What inequality models “at least $40$ grams of protein” if foods give $6x$ and $9y$ grams?
Tap to reveal answer
$6x + 9y \ge 40$. Nutritional requirement must be satisfied.
$6x + 9y \ge 40$. Nutritional requirement must be satisfied.
← Didn't Know|Knew It →