California High School Proficiency Examination (CHSPE) Mathematics: Algebra and Patterns Syllabus

Solve the two-step equation x/5 - 2 = 3.

Add 2: x/5 = 5; then multiply both sides by 5: x = 25.

What is the general strategy for solving a multi-step equation?

1) Distribute to remove parentheses, 2) combine like terms on each side, 3) move variable terms to one side and constants to the other, 4) isolate the variable using inverse operations.

Solve 3(x - 2) = 12.

Distribute: 3x - 6 = 12; add 6: 3x = 18; divide by 3: x = 6.

Solve the equation with variables on both sides: 5x - 4 = 2x + 11.

Subtract 2x: 3x - 4 = 11; add 4: 3x = 15; divide by 3: x = 5.

When solving an equation, what does it mean if you end with a false statement like 3 = 7?

It means the equation has no solution (the variable cancels out and the remaining statement is never true).

When solving an equation, what does it mean if you end with a true statement like 5 = 5?

It means the equation has infinitely many solutions (it is an identity, true for all values of the variable).

What are the four basic inequality symbols and their meanings?

< less than, > greater than, ≤ less than or equal to, ≥ greater than or equal to.

What is the one special rule when solving inequalities that does not apply to equations?

When you multiply or divide both sides by a negative number, you must reverse (flip) the direction of the inequality symbol.

Solve the inequality -2x > 6.

Divide both sides by -2 and flip the symbol: x < -3.

On a number line, how do you graph x ≤ 4 versus x < 4?

For x ≤ 4 use a closed (filled) circle at 4; for x < 4 use an open circle at 4. Both shade to the left toward smaller values.

When graphing an inequality on a number line, when do you use an open versus a closed circle?

Open circle for strict inequalities (< or >) because the endpoint is not included; closed (filled) circle for ≤ or ≥ because the endpoint is included.

What are the general steps to solve a word problem using an equation?

1) Read and identify the unknown, 2) define a variable for it, 3) translate the words into an equation, 4) solve the equation, 5) check the answer against the problem's conditions.

A number tripled and increased by 4 equals 19. Write and solve the equation.

3x + 4 = 19; subtract 4: 3x = 15; divide by 3: x = 5.

What is the coordinate plane and what are its main parts?

A flat surface formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. They cross at the origin (0,0) and divide the plane into four quadrants.

What does an ordered pair (x, y) represent on the coordinate plane?

The location of a point: x is the horizontal distance from the origin, y is the vertical distance. Order matters—x comes first, y second.

In which quadrant are both coordinates positive, and how are the four quadrants numbered?

Quadrant I (top right) has (+, +). Numbering goes counterclockwise: I (+,+), II (-,+), III (-,-), IV (+,-).

What are the coordinates of the origin?

(0, 0) — the point where the x-axis and y-axis intersect.

What is the slope-intercept form of a linear equation and what does each letter mean?

y = mx + b, where m is the slope (rate of change) and b is the y-intercept (where the line crosses the y-axis).

What is the formula for the slope of a line through points (x1, y1) and (x2, y2)?

m = (y2 - y1) / (x2 - x1), the change in y divided by the change in x ('rise over run').

How do you graph a line given in slope-intercept form y = mx + b?

Plot the y-intercept (0, b) first, then use the slope m as rise/run to find a second point, and draw a straight line through them.

What do positive, negative, zero, and undefined slopes look like?

Positive slope rises left to right; negative slope falls left to right; zero slope is a horizontal line; an undefined slope is a vertical line.

What is the x-intercept and the y-intercept of a graph?

The x-intercept is where the line crosses the x-axis (y = 0); the y-intercept is where the line crosses the y-axis (x = 0).

When interpreting a real-world line graph, what does the slope represent?

The rate of change—how much the quantity on the y-axis changes for each one-unit increase in the quantity on the x-axis (e.g., miles per hour, dollars per item).

On a distance-versus-time graph, what does a horizontal (flat) segment indicate?

No change in distance over time—the object is stopped/at rest.

What is a function?

A relation in which each input (x-value) is paired with exactly one output (y-value).

What is the vertical line test and what does it determine?

If any vertical line drawn through a graph crosses it more than once, the graph is NOT a function; if every vertical line touches the graph at most once, it IS a function.

What are the domain and range of a function?

The domain is the set of all input values (x-values); the range is the set of all output values (y-values).

In function notation f(x) = 2x + 1, what does f(3) mean and equal?

f(3) means substitute x = 3 into the function: f(3) = 2(3) + 1 = 7.

What is the difference between an arithmetic pattern and a geometric pattern?

An arithmetic pattern changes by adding/subtracting the same constant (common difference) each step (e.g., 2,5,8,11). A geometric pattern changes by multiplying/dividing by the same constant (common ratio) each step (e.g., 3,6,12,24).

In the sequence 4, 9, 14, 19, ..., what is the common difference and the next term, and what is the rule for the nth term?

The common difference is 5; the next term is 24. The nth term rule is 5n - 1 (since the first term is 4 = 5(1) - 1).