Math: Lines, Inequalities and Systems

This is Unit Nine of ACT Preparation: lines, inequalities and systems. Unit eight kept every rewrite equal in value; this unit uses that skill to solve equations and to describe lines. One idea runs through it: a line is a rate of change and a starting value, and every form of its equation says the same two things. By the end of this part you will be able to find intercepts, distances and midpoints. To solve linear equations, including those with fractions. To turn a word problem into an equation. To write a line in any form and read what its slope means. And to solve absolute value equations and inequalities.

Three entries on the coordinate plane: intercepts, distance and midpoint.

Points and intercepts.

ACT 0801 The Coordinate Plane and Intercepts

A point is an ordered pair, x then y. The x-intercept is where a graph crosses the x-axis, found by setting y to zero; the y-intercept, by setting x to zero. For 2x plus 3y equals twelve, the intercepts are six and four. An intercept is a point: six, zero, not six alone.

The distance between two points.

ACT 0802 The Distance Formula

The distance between two points is the square root of the horizontal gap squared plus the vertical gap squared: the Pythagorean theorem on the grid. From one, two to four, six, the gaps are three and four, so the distance is five. Square before adding, and add before taking the root.

The point halfway between.

ACT 0803 The Midpoint Formula

The midpoint of a segment averages the endpoints' coordinates: between two, negative three and eight, five, the midpoint is five, one. Given a midpoint and one endpoint, the other lies as far past the midpoint again.

Five entries on solving equations and writing them from words.

Solving a linear equation.

ACT 0804 Solving a Linear Equation

A linear equation is solved by inverse operations: distribute, combine like terms, gather the variable on one side, then divide by its coefficient. Three times x minus two equals x plus eight becomes 2x equals fourteen, so x is seven. Check by substituting: both sides give fifteen.

Equations with fractions, step by step.

ACT 0805 Solving an Equation With Fractions

Find the least common denominator, multiply every term on both sides by it, solve, and reject any answer that makes a variable denominator zero. X over three plus x over four equals seven becomes 4x plus 3x equals eighty-four, so x is twelve. Two over x minus one equals six over x plus three gives x equals three, which makes neither denominator zero. Multiply every term, including terms with no fraction: in x over two plus three equals five, the three becomes six.

Solving a formula for one letter.

ACT 0806 Literal Equations

A literal equation, such as a formula, is solved for one variable in terms of the others, by the same inverse operations. Solving d equals r t for t gives d over r. Treat every other letter as a fixed number while isolating the one you want.

Turning words into symbols.

ACT 0807 Translating Words Into Equations

To write an equation from words, name the unknown, then translate phrase by phrase: is means equals, of often means times, and less than reverses the order. "Five less than twice a number is seventeen" is 2n minus five equals seventeen, not five minus 2n.

A word problem, step by step.

ACT 0808 Writing a Word Problem as an Equation

Name what the question asks for with a variable and its units, express every other quantity in terms of it, find the fact that makes two expressions equal, then solve and answer the question asked. Two trains leave in opposite directions at sixty and seventy-five miles per hour; together they cover 135 miles each hour, so they are 405 miles apart after three hours. A question asking for the faster train's distance wants 225 miles, not three.

Nine entries on lines: slope, the forms of a line, and what a model's numbers mean.

Rise over run.

ACT 0809 Slope

The slope of a line is rise over run: the change in y divided by the change in x, the rate of change of y per unit of x. Through one, three and four, nine, the slope is six over three, two. Subtract in the same order, top and bottom. Positive slope rises left to right.

Slope-intercept form.

ACT 0810 Slope-Intercept Form

In slope-intercept form, y equals m x plus b, m is the slope and b the y-intercept. Y equals negative 3x plus five crosses the y-axis at five and falls three for each step right. Solve for y first: 4x plus 2y equals six becomes y equals negative 2x plus three, slope negative two.

Two more forms of a line.

ACT 0811 Point-Slope and Standard Form

Point-slope form builds a line from a point and a slope: y minus y one equals m times x minus x one. Standard form is A x plus B y equals C. Slope two through three, one gives y minus one equals two times x minus three, or y equals 2x minus five. A negative coordinate turns into a plus sign inside the parentheses.

Flat and upright lines.

ACT 0812 Horizontal and Vertical Lines

A horizontal line, y equals a constant, has slope zero. A vertical line, x equals a constant, has an undefined slope, because its run is zero. Zero slope and undefined slope are different: zero over four is zero, but four over zero is undefined.

The line through two points, step by step.

ACT 0813 The Line Through Two Points

Find the slope from the two points, put it with either point into point-slope form, then rearrange and check the other point. Through negative one, four and three, negative four, the slope is negative eight over four, negative two, and the line is y equals negative 2x plus two; the point three, negative four fits. If the two points lead to different equations, look for a sign error from subtracting in different orders.

Parallel and perpendicular slopes.

ACT 0814 Parallel and Perpendicular Slopes

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals, with a product of negative one, unless one is vertical and the other horizontal. For slope two thirds, a perpendicular slope is negative three halves. Flip and change the sign; doing only one gives a wrong slope.

The perpendicular line through a point, step by step.

ACT 0815 The Perpendicular Line Through a Point

Put the given line in slope-intercept form and read its slope, take the negative reciprocal, then use point-slope form with the given point. For the line through four, one perpendicular to 2x minus 3y equals six, the given slope is two thirds, the new slope negative three halves, and the line y equals negative three halves x plus seven. Read the slope only after solving for y: the coefficient two in standard form is not the slope.

What slope and intercept mean.

ACT 0816 Slope and Intercept in a Model

In a linear model, the slope is the rate of change, with units, and the intercept is the starting value, when the input is zero. For a plumber's bill of forty-five plus thirty times the hours, forty-five dollars is the visit fee and thirty dollars is the charge per hour. The slope's units are dollars per hour.

Interpreting a model, step by step.

ACT 0817 Interpreting a Linear Model

Identify the input and output with units, read the slope as the change per unit of input and the intercept as the output at zero, and answer in the language of the situation. A tank's level is 800 minus 25t liters: it falls twenty-five liters a minute from 800, and it runs dry at thirty-two minutes, the t-intercept. If the input cannot be zero in the situation, the intercept is only where the line starts.

Four entries on absolute value and inequalities.

Absolute value as distance.

ACT 0818 Absolute Value Equations

The absolute value of x is its distance from zero. An equation setting the absolute value of x minus a equal to a positive b splits into two: x minus a equals b, or negative b. The absolute value of x minus three equals five gives eight or negative two. If b is negative, there is no solution.

The one exception for inequalities.

ACT 0819 Inequalities: the Sign Flip

A linear inequality is solved like an equation, with one exception: multiplying or dividing by a negative reverses the sign. Negative 3x plus four greater than nineteen gives negative 3x greater than fifteen, so x is less than negative five. Test a value: negative six gives twenty-two, greater than nineteen.

Two conditions at once.

ACT 0820 Compound Inequalities

A compound inequality joins two conditions: with and, both hold; with or, either may. Apply each step to all three parts: negative one is at most 2x plus three, which is less than nine, gives x from negative two up to but not including three. The bracket includes negative two; the parenthesis excludes three.

Absolute value inequalities.

ACT 0821 Absolute Value Inequalities

An absolute value less than b means within b: a single interval, between. Greater than b means farther than b: two pieces, outside. The absolute value of x minus four less than three gives one to seven; greater than three gives below one or above seven.

Look at what you can now do. You can find intercepts, distances and midpoints, writing an intercept as a point. You can solve equations by inverse operations, multiplying every term to clear fractions. You can turn a word problem into an equation and answer the question it asked. You can find a slope, write a line in any form, and read its slope and intercept as a rate and a starting value. And you can solve absolute value equations and inequalities, flipping the sign for a negative. Part two solves inequalities step by step, then turns to systems.