Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 3 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Count solutions by counting intersections
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.
Solutions to the equation correspond to shared graph points, not to the number of curves drawn. We’ll trace the two complete lines and count each distinct place where they have …
Preview problemUse sign changes to bracket an approximate solution
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.
Equality occurs where the difference between the functions is zero. We’ll scan the table for adjacent entries with opposite signs and use the closest such pair as the narrowest supported …
Preview problemSolve a break-even problem by setting models equal
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.
Break-even means the two complete cost models have equal outputs for the same month. We’ll build each model from its fixed charge and monthly rate, set the totals equal, and …
Preview problemInterpret technology-reported intersection coordinates
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.
Technology reports coordinates, but the context supplies their meaning. We’ll map the first coordinate to the horizontal variable and its unit, map the second to the output variable and its …
Preview problemGraph one linear inequality in slope-intercept form
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
A linear inequality graph combines a boundary decision with a half-plane decision. We’ll replace the inequality sign by equality to construct the line from its slope and intercept, use the …
Preview problemGraph a standard-form linear inequality
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
Standard form still describes one boundary line and one solution half-plane. We’ll replace the inequality sign by equality, rearrange carefully to expose the line’s signed slope, and find both axis …
Preview problemUse a test point to decide shading
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
A test point decides between the two sides of a boundary only after substitution. We’ll place its y-coordinate on the left, use its x-coordinate in the entire expression on the …
Preview problemGraph a system of two linear inequalities
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
A system graph keeps only points satisfying both inequalities at once. We’ll construct each boundary separately, use each inclusive symbol to set its style, and identify the required half-plane for …
Preview problemCheck whether a point is in a solution region
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
A point belongs to a system’s solution region only if it satisfies every constraint at once. We’ll substitute its coordinates into each inequality separately, preserve the ordered-pair roles, and evaluate …
Preview problemWrite inequalities from a graphed region
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
A shaded graph encodes three pieces for each inequality: the boundary equation, whether the boundary is included, and which side is allowed. We’ll read the slanted and horizontal boundaries separately, …
Preview problemModel a feasible region from a context
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
The context supplies four distinct constraints, and each must keep its original variable meaning. We’ll attach each time coefficient to the matching poster count, translate the maximum time and minimum …
Preview problemInterpret a feature of a feasible-region graph
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
The boundary should be interpreted by connecting each algebraic piece back to the context. We’ll turn each item count into its resource contribution, determine what total the boundary equation fixes, …
Preview problemChoose solid or dashed boundary for an inequality
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
Boundary style comes entirely from whether equality is allowed. We’ll replace the inequality sign with equality to identify the boundary, inspect the original symbol for an equality bar, and map …
Preview problemConvert a boundary and shaded side into an inequality
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
The graph must be decoded in layers. We’ll use its intercepts to determine the boundary’s slope and equation, use the line style to decide whether equality belongs, and test a …
Preview problemFind the overlap of three inequalities
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
The overlap must satisfy all three half-plane conditions simultaneously. We’ll use the two coordinate restrictions to locate the allowed quadrant, graph the remaining boundary from its intercepts, and keep every …
Preview problemDecide whether a boundary point is included
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
A point on a boundary is not automatically included or excluded. We’ll substitute both coordinates into the named constraint, compare the two sides exactly, and then inspect whether the inequality …
Preview problemSolve a one- or two-step linear equation
Solve linear equations and inequalities in one variable, including literal linear equations.
The variable is wrapped first by multiplication and then by addition, so the inverse operations must undo those layers in reverse order. We’ll apply each operation to both sides, isolate …
Preview problemSolve a linear equation in one variable by distributing and using inverse operations
Solve linear equations and inequalities in one variable, including literal linear equations.
The outside factor applies to the entire grouped expression. We’ll distribute it to every term, simplify the equivalent linear equation, and then undo the remaining addition and multiplication with balanced …
Preview problemSolve a linear equation with variables on both sides and classify the result
Solve linear equations and inequalities in one variable, including literal linear equations.
Having the variable on both sides does not determine the equation’s classification by itself. We’ll collect variable terms on one side and constants on the other using balanced operations, then …
Preview problemSolve and graph a one-variable linear inequality
Solve linear equations and inequalities in one variable, including literal linear equations.
Solving gives both a boundary value and a direction for the entire solution set. We’ll isolate the variable with balanced operations, checking the sign of every multiplier or divisor to …
Preview problemSolve a compound inequality
Solve linear equations and inequalities in one variable, including literal linear equations.
A chained inequality requires both comparisons to hold at the same time, so its solution is an overlap between two bounds. We’ll perform the same inverse operation on all three …
Preview problemSolve a literal linear equation for a specified variable
Solve linear equations and inequalities in one variable, including literal linear equations.
A literal equation is solved with the same balance principle as a numerical equation; the other letters simply act as given quantities. We’ll undo the addition around the target variable …
Preview problemSolve and interpret a context inequality
Solve linear equations and inequalities in one variable, including literal linear equations.
The budget limits a total cost, while the variable counts objects rather than arbitrary real numbers. We’ll multiply the per-ticket price by the count, translate the upper-limit language into an …
Preview problemSolve a linear equation by clearing denominators
Solve linear equations and inequalities in one variable, including literal linear equations.
Fractions can be removed without changing an equation’s solutions by multiplying the entire equation by a common denominator. We’ll find the least common denominator, apply it to every term—including the …
Preview problemSolve a linear inequality with fractions or decimals
Solve linear equations and inequalities in one variable, including literal linear equations.
A decimal coefficient changes the arithmetic, but not the logic of isolating the variable. We’ll remove the added constant, interpret the decimal as a familiar fraction, and divide both sides …
Preview problemSolve a linear equation after combining like terms on both sides
Solve linear equations and inequalities in one variable, including literal linear equations.
Simplifying first makes the equation easier to solve and reduces sign errors. We’ll combine only terms with the same variable part, keep the constant separate, and then isolate the resulting …
Preview problemClassify a linear equation as one solution, no solution, or infinitely many solutions
Solve linear equations and inequalities in one variable, including literal linear equations.
When matching variable terms appear on both sides, canceling them is only the beginning of the classification. We’ll subtract the shared term with a balanced operation and inspect the constant …
Preview problemTranslate between inequality, interval notation, and number-line graph
Solve linear equations and inequalities in one variable, including literal linear equations.
Inequality notation, interval notation, and a number-line graph must preserve the same three facts: boundary, inclusion, and direction. We’ll read strictness from the inequality symbol to set the endpoint marker …
Preview problemSolve a linear inequality in one variable using inverse operations when dividing by a negative flips the inequality
Solve linear equations and inequalities in one variable, including literal linear equations.
Isolating the variable requires dividing by its coefficient, and the coefficient’s sign controls the inequality direction. We’ll divide both sides by the same negative number and reverse the comparison because …
Preview problemSolve an absolute-value equation by splitting it into two linear cases
Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.
Absolute value describes distance, so a positive target places possible values on both sides of a center. We’ll set the expression inside the bars equal to the positive target and …
Preview problemSolve an absolute-value equation after isolating the absolute value
Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.
The absolute-value expression must be isolated before its two distance cases are created. We’ll first undo the multiplication outside the bars, then equate the inside expression to both signed versions …
Preview problemSolve an absolute-value inequality with less than or less than or equal to
Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.
An upper bound on absolute value keeps the variable within a fixed distance of the center, so the solution lies between two boundaries. We’ll rewrite that distance condition as one …
Preview problemSolve an absolute-value inequality with greater than or greater than or equal to
Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.
A lower bound on absolute value asks for points at least a fixed distance from the center, so the solution occupies two outside regions. We’ll translate that condition into a …
Preview problemGraph an interval solution on a number line
Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.
A compound inequality specifies one bounded set, but its two endpoints can have different inclusion rules. We’ll read each comparison separately to assign an open or closed marker, then shade …
Preview problemInterpret an absolute-value statement in context
Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.
In context, the expression inside absolute value compares an actual measurement with its target, and the bars turn that difference into distance. We’ll identify the center, read the comparison as …
Preview problemDecide whether an absolute-value equation with a negative target is possible
Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.
Before splitting an absolute-value equation into cases, compare its target with the possible outputs of absolute value. We’ll use the distance interpretation to establish that range, inspect the target’s sign, …
Preview problem