Math: Lines, Inequalities and Systems

This is Unit Nine, part two, of ACT Preparation. Part one closed on absolute value inequalities; this part solves inequalities step by step, then turns to systems, where two lines meet. The governing idea carries over: a line is a rate of change and a starting value. A system asks where two such lines agree. By the end you will be able to solve linear and absolute value inequalities. To tell how many solutions a system has. To solve a system by substitution or elimination. To write and solve a two-unknown word problem. And to check points against a system of inequalities.

Two entries that solve inequalities step by step.

A linear inequality, step by step.

ACT 0822 Solving a Linear Inequality

Distribute and combine, gather the variable on one side, divide by its coefficient, reversing the sign if it is negative, and test one value inside the solution and one outside. Five minus two times x plus four is at least 3x plus seven becomes negative 5x at least ten, and dividing by negative five gives x at most negative two. Negative three tests true and zero tests false. Gathering the variable where its coefficient stays positive avoids the flip altogether, with the same answer.

An absolute value inequality, step by step.

ACT 0823 Solving an Absolute Value Inequality

Isolate the absolute value, decide its type, less than for one interval between or greater than for two pieces outside, write the compound inequality without bars, and test each region. Two times the absolute value of x plus one, minus three, at most seven, becomes the absolute value of x plus one at most five, so x runs from negative six to four. Zero tests true and five tests false. Isolate before splitting; splitting with the two and the three still outside the bars gives wrong bounds.

Seven entries on systems: what a solution is, how many there can be, and two ways to find one.

What solves a system.

ACT 0824 What a System's Solution Is

A solution of a system is an ordered pair that satisfies every equation at once; on a graph, it is where the lines intersect. Two, three solves x plus y equals five and 2x minus y equals one, because both hold there. A point on only one line is not a solution.

How many solutions.

ACT 0825 One, None or Infinitely Many

Different slopes give one solution. The same slope with different intercepts gives parallel lines, no solution, and solving gives a false statement like zero equals five. The same line twice gives infinitely many, and an identity like zero equals zero. Compare the slopes first.

Substitution.

ACT 0826 Substitution

Substitution solves one equation for one variable and substitutes that expression into the other equation. With y equals x plus two and 3x plus y equals fourteen, 3x plus x plus two equals fourteen, so x is three and y is five. Substitute into the other equation; using the same one gives an identity.

Elimination.

ACT 0827 Elimination

Elimination adds or subtracts the equations so one variable cancels, after scaling so its coefficients are opposites. X plus 2y equals seven and 3x minus 2y equals five add to 4x equals twelve, so x is three and y is two. Multiply every term when scaling, including the constant.

Substitution, step by step.

ACT 0828 Solving a System by Substitution

Solve one equation for a variable that is isolated or has coefficient one, substitute into the other, solve, then back-substitute and check both. From 2x plus y equals eleven, y is eleven minus 2x; then 5x minus three times that quantity equals twenty-two gives 11x equals fifty-five, so x is five and y is one. Wrap the substituted expression in parentheses; without them, the negative three multiplies only the eleven.

Elimination, step by step.

ACT 0829 Solving a System by Elimination

Line both equations up, multiply so one variable's coefficients are opposites, add to remove it, then substitute back and check. For 3x plus 4y equals ten and 2x minus 3y equals one, multiplying by two and by negative three gives 17y equals seventeen, so y is one and x is two. Choose the variable that needs the smaller multipliers.

Two unknowns from a story.

ACT 0830 A Two-Unknown Word Problem

Name both unknowns, write one equation for each fact the story gives, such as a count and a total value, solve the system, and check both facts. A theater sold 300 tickets, adults at $12 and students at $8, for $3,040: a plus s equals 300 and 12a plus 8s equals 3,040 give 160 adult tickets and 140 student tickets. The two equations must describe different facts; the count written twice gives no single answer.

Four entries on inequality systems, curves, and the whole unit in one problem.

Graphing a system of inequalities.

ACT 0831 Systems of Inequalities

Draw each boundary line, solid if equality is included and dashed if not, shade each solution side, and keep the overlap. Y at least x and y less than four overlap above the line y equals x and below a dashed y equals four. Choose each side by testing a point off the boundary.

Checking a point, step by step.

ACT 0832 Which Points Satisfy a System of Inequalities

Substitute the point into each inequality, decide whether each is true with attention to the boundary, and keep the point only if every inequality holds. Against y greater than 2x minus one and x plus y at most six, the point one, three is in; three, four fails the first and is out; and two, four lies in the region, on the solid boundary of the second. A point on a dashed boundary is outside; the strict sign decides.

Where a line meets a curve.

ACT 0833 A Line Meets a Curve

A line can meet a parabola or a circle in zero, one or two points. Substituting the line into the curve gives a quadratic, whose solutions count the intersections: y equals x meets y equals x squared at zero, zero and one, one. Each x found gives its own point.

The whole unit in one problem.

ACT 0834 Two Plans, One Crossing

A line is a rate of change and a starting value, and a system asks where two lines agree. Plan A charges $20 a month plus ten cents a minute; plan B charges $35 plus five cents a minute. The slopes are dollars per minute and the intercepts the monthly fees. The plans cost the same where the lines cross: twenty plus 0.1m equals thirty-five plus 0.05m gives 300 minutes, at $50 each. Below 300 minutes plan A wins; above it, plan B, and the inequality agrees: B is cheaper when m exceeds 300. Read the slope only after solving for y, check the sign whenever you divide by a negative, and interpret the crossing by examining each side of it.

Look at what you can now do. You can solve a linear inequality and an absolute value inequality, testing a value in each region. You can tell from the slopes whether a system has one solution, none, or infinitely many. You can solve a system by substitution, with parentheses, or by elimination, with the smaller multipliers. You can turn a story with two unknowns into two equations about different facts. And you can check points against a system of inequalities, and find where a line meets a curve. Unit ten generalizes the line into the function.