If you have a week, read it top to bottom, and work every example on paper before you read its solution. If you have an evening, go straight to Part 5 and Part 6, then do the self-check at the end, and come back to whichever parts your results point to. Each formula is marked to say whether the test hands it to you on the formula sheet or you carry it in. Everything here is what these questions look like on the test; nothing is included that the test does not ask.
The grid: where a point lives
If the grid is new to you, start with The Coordinate Plane, which builds it one step at a time. This part is a short review.
The coordinate plane is two number lines crossing at right angles. The one lying flat is the x-axis; the one standing up is the y-axis. They cross at the origin, the point (0, 0).
A point is named by two numbers in parentheses, always in the same order: (x, y). The first number says how far to go across from the origin; the second says how far to go up or down. Positive x is to the right, negative x is to the left. Positive y is up, negative y is down.
The axes cut the plane into four quadrants, numbered with Roman numerals, counterclockwise from the upper right. In quadrant I both numbers are positive. In II, x is negative and y is positive. In III both are negative. In IV, x is positive and y is negative. A point that sits on an axis is in no quadrant: a point on the x-axis has y = 0, like (5, 0), and a point on the y-axis has x = 0, like (0, −3).
Read the four points marked on the grid.
Start at the origin every time. A: 3 to the right, then 2 up, so A is (3, 2). B: 4 to the left, then 1 up: (−4, 1). C: 2 to the left, then 3 down: (−2, −3). D: 4 to the right, then 2 down: (4, −2).
The order matters. (2, 3) is a different point from (3, 2): the first is 2 across and 3 up, the second is 3 across and 2 up. When a point has a negative number, the sign tells you the direction, left instead of right or down instead of up; it does not change which number comes first.
Distance and midpoint
When two points sit on the same horizontal line or the same vertical line, the distance between them is a subtraction, or a count of squares. From (2, 3) to (7, 3) is 7 − 2 = 5. From (4, −1) to (4, 6) is 6 − (−1) = 7: subtracting a negative adds, and you can see the 7 by counting from −1 up to 6.
When the two points are not lined up, draw the right triangle between them. The distance across is one leg, the distance up or down is the other, and the straight line between the points is the hypotenuse. The Pythagorean theorem, which is on the formula sheet, gives the hypotenuse from the two legs:
How far is it from A (1, 1) to B (7, 9)?
Across: 7 − 1 = 6. Up: 9 − 1 = 8. Those are the legs, so a = 6 and b = 8.
a² + b² = 6² + 8² = 36 + 64 = 100, and c = √100 = 10. The distance is 10.
A distance is always positive. If you subtract in the other order and get −6 and −8, squaring removes the signs, and the answer is the same 10.
If you have met the distance formula, d = √((x₂ − x₁)² + (y₂ − y₁)²), it is this same triangle written out in one line: the two subtractions are the legs. It is not on the formula sheet; the Pythagorean theorem is, and it does the same job.
The midpoint
The midpoint is the point halfway along the segment between two points. Halfway between two numbers is their average: add them and divide by 2. The midpoint does that twice, once for the x values and once for the y values.
What is the midpoint of the segment from A (−5, 3) to B (3, −1)?
Average the x values: (−5 + 3) ÷ 2 = −2 ÷ 2 = −1.
Average the y values: (3 + (−1)) ÷ 2 = 2 ÷ 2 = 1.
The midpoint is (−1, 1). On the grid it sits halfway along the segment, 4 squares from each end, which is a good way to check.
Slope
Slope measures how steep a line is and which way it leans. It is the rise over the run: how much the line goes up or down for each step it goes across.
The small 1s and 2s only mean "the first point" and "the second point." Subtract the y values on top and the x values on the bottom, in the same order both times: if you start from the second point on top, start from the second point on the bottom too.
Four kinds of slope
A line that goes up to the right has a positive slope. A line that goes down to the right has a negative slope. A horizontal line has slope 0: it never rises, so the top of the fraction is 0. A vertical line has no slope at all, called undefined: it never runs, so the bottom of the fraction is 0, and division by 0 has no answer. The steeper the line, the larger the slope, ignoring the sign: a slope of 3 is steeper than a slope of 12, and a slope of −3 is just as steep as 3, leaning the other way.
Slope from a graph
Pick two points where the line passes exactly through a corner of the grid. Count how far up from the lower point to the level of the higher one (the rise), count how far across (the run), and divide.
What is the slope of the line on the grid?
The line passes cleanly through (1, 2) and (4, 8). From (1, 2) up to the level of (4, 8) is a rise of 8 − 2 = 6. Across from 1 to 4 is a run of 4 − 1 = 3.
Slope = 63 = 2. The line climbs 2 for every 1 it moves to the right, and you can see that too: from (1, 2), one step right and two steps up lands on (2, 4), which is on the line.
Slope from two points
What is the slope of the line through (−1, 4) and (3, −2)?
Call (−1, 4) the first point and (3, −2) the second. Rise: y₂ − y₁ = −2 − 4 = −6. Run: x₂ − x₁ = 3 − (−1) = 3 + 1 = 4.
Slope = −64 = −32. The line drops 3 for every 2 it moves to the right; the negative sign says it goes down.
Taking the points in the other order gives (4 − (−2)) over (−1 − 3) = 6 over −4, which is the same −32. Either order works, as long as you keep the same order on the top and the bottom.
Slope from a table
If y changes by the same amount every time x goes up by 1, the points in the table all lie on one straight line. That amount is the slope. Here each step of 1 in x adds 3 to y, so the slope is 3; and the y value at x = 0 is 5, which is where the line crosses the y-axis. The rule is y = 3x + 5. If x steps by 2 instead of 1, divide the change in y by 2.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 |
Slope as a rate
In a word problem, the slope is a rate: something per something. A plumber charges $60 to come out, plus $45 an hour, so the cost of h hours is 45h + 60. The 45 is the slope: each extra hour adds $45 to the bill. The 60 is where the line starts, the y-intercept (Part 4). The test calls the slope the rate of change when it asks about a graph of a real situation: dollars per hour, miles per gallon, gallons per minute.
Parallel and perpendicular
Parallel lines never meet, because they have the same slope. Perpendicular lines cross at a right angle, and their slopes are negative reciprocals of each other: flip the fraction and change the sign. A line with slope 2 is perpendicular to a line with slope −12; a line with slope −34 is perpendicular to one with slope 43. The test asks about this less often than the rest of this page. When it does, the rule above is all you need.
The two intercepts
An intercept is where a line crosses an axis. The y-intercept is where the line crosses the y-axis. Every point on the y-axis has x = 0, so to find the y-intercept, put 0 in place of x and see what y comes out. The x-intercept is where the line crosses the x-axis. Every point there has y = 0, so put 0 in place of y and solve for x.
A y-intercept can be written as a point, (0, 3), or as the single number 3. The test uses both, and "the y-intercept is 3" means the point (0, 3). The same goes for an x-intercept of 4, which is the point (4, 0).
Find both intercepts of the line 3x + 4y = 12.
y-intercept: put 0 in for x. 3(0) + 4y = 12, so 4y = 12 and y = 3. The line crosses the y-axis at (0, 3).
x-intercept: put 0 in for y. 3x + 4(0) = 12, so 3x = 12 and x = 4. The line crosses the x-axis at (4, 0).
Two points make a line, so the two intercepts are the fastest way to graph an equation written like this: mark (0, 3), mark (4, 0), and draw the line through them. The grid above is exactly that.
What the intercepts mean in a story
When a line describes a real situation, the y-intercept is the starting value, what y is before anything has happened: the plumber's $60 fee before any hours, the water in a tank before the drain opens. The x-intercept is when y reaches zero: the minute the tank is empty, the month a loan is paid off.
A tank holds 300 gallons. It drains at 20 gallons a minute, so the water left after x minutes is y = 300 − 20x. What are the intercepts, and what do they mean?
y-intercept: x = 0 gives y = 300. Before the drain opens, the tank holds 300 gallons.
x-intercept: set y = 0: 0 = 300 − 20x. Add 20x to both sides: 20x = 300, so x = 15. The tank is empty after 15 minutes.
The slope, −20, is the rate: the tank loses 20 gallons every minute, which is why it is negative.
The three forms of a linear equation
A linear equation is one whose graph is a straight line. The same line can be written in three ways, and each way makes something easy to see. The test uses all three and asks you to move between them, so the skill is not only to recognize each form but to turn one into another.
| Form | What it looks like | What you can read off it | Best for |
|---|---|---|---|
| Slope-intercept | y = mx + b | the slope m and the y-intercept b | graphing quickly; reading the slope and the starting value; comparing two lines |
| Point-slope | y − y₁ = m(x − x₁) | the slope m and one point (x₁, y₁) on the line | writing the equation when you are given a point and the slope, or two points |
| Standard | Ax + By = C | nothing directly; the intercepts come out in one step each | finding the intercepts; the way many test questions state a line |
1. Slope-intercept form
The number multiplying x is the slope; the number added at the end is the y-intercept. Read both straight off. For y = −3x + 5, the slope is −3 and the y-intercept is 5, the point (0, 5). Watch the sign: y = 2x − 7 has a y-intercept of −7, not 7.
To graph it: mark the y-intercept on the y-axis. Then use the slope to reach a second point. A slope of −3 is −31, so from (0, 5) go down 3 and right 1 to (1, 2). Draw the line through the two points.
2. Point-slope form
Use it when you know the slope and one point on the line, or when you know two points, since two points give you the slope. Put the slope in for m and the point in for x₁ and y₁, and the equation is written. To get slope-intercept form from it, multiply out the right side and move the number away from y.
Write the equation of the line with slope 2 that passes through (3, 1).
m = 2, x₁ = 3, y₁ = 1. Point-slope form: y − 1 = 2(x − 3). That already is an equation of the line, and a test answer might be written this way.
To get slope-intercept form, multiply out the right side: y − 1 = 2x − 6. Then add 1 to both sides: y = 2x − 5.
Check with the point: x = 3 gives 2(3) − 5 = 1. It lands on (3, 1), as it should.
Write the equation of the line through (1, 4) and (3, 10).
First the slope: rise over run = (10 − 4) over (3 − 1) = 62 = 3.
Now point-slope form with either point. Using (1, 4): y − 4 = 3(x − 1). Multiply out: y − 4 = 3x − 3. Add 4 to both sides: y = 3x + 1.
Check with the other point: x = 3 gives 3(3) + 1 = 10, and the point was (3, 10). Using (3, 10) instead would give y − 10 = 3(x − 3), which works out to the same y = 3x + 1.
3. Standard form
Both variables sit on one side and a plain number on the other; A, B, and C are whole numbers, and A is usually written positive. The intercepts are quick from this form (Part 4). The slope is not visible; to read it, solve for y, which turns the equation into slope-intercept form.
Find the slope and y-intercept of 2x + 3y = 6.
Solve for y. Take 2x from both sides: 3y = −2x + 6. Divide every term by 3: y = −23x + 2.
Now it is slope-intercept form: the slope is −23 and the y-intercept is 2. Check with the x-intercept: y = 0 gives 2x = 6, x = 3, and −23(3) + 2 = −2 + 2 = 0. The point (3, 0) is on the line.
Write 4x − 2y = 6 in slope-intercept form.
Take 4x from both sides: −2y = −4x + 6. Divide every term by −2, and remember that dividing by a negative changes every sign: y = 2x − 3.
The slope is 2 and the y-intercept is −3. A quick check: x = 0 gives y = −3 in the new form, and in the original 4(0) − 2(−3) = 6. Both agree.
Two lines that break the pattern
A horizontal line has an equation like y = 3: every point on it has y = 3, whatever x is. Its slope is 0; in slope-intercept form it is y = 0x + 3. A vertical line has an equation like x = −2: every point on it has x = −2. Its slope is undefined, and it cannot be written as y = mx + b at all. If a test question shows a vertical or horizontal line, these are the equations it is looking for.
Matching an equation to a graph
Look at where the line crosses the y-axis, and whether it goes up or down to the right. That rules out most wrong choices. Then test one clear point: a point on the line makes the equation true when you put its x and y in.
A graph shows a line through (0, −1) and (2, 3). Which equation is it: y = 2x − 1, y = 2x + 1, or y = −2x − 1?
The line crosses the y-axis at (0, −1), so b = −1. That leaves y = 2x − 1 and y = −2x − 1. The line goes up to the right (from (0, −1) up to (2, 3)), so the slope is positive: y = 2x − 1.
Confirm with the second point: x = 2 gives 2(2) − 1 = 3. The point (2, 3) is on the line. Or compute the slope directly: (3 − (−1)) over (2 − 0) = 4 over 2 = 2.
Solving a linear equation
An equation is a balance. The two sides weigh the same, and whatever you do to one side you must do to the other, or it stops being true. To solve for x means to get x alone on one side. You do that by undoing what has been done to x, in reverse order: undo the adding and subtracting first, then the multiplying and dividing. Every step, ask: what is still attached to x, and what undoes it?
Solve 3x + 7 = 22.
x has been multiplied by 3, then 7 was added. Undo the adding first: take 7 from both sides. 3x = 15.
Now undo the multiplying: divide both sides by 3. x = 5.
Check by putting 5 back in: 3(5) + 7 = 15 + 7 = 22. It balances.
Solve 5x − 4 = 2x + 14.
There are x terms on both sides. Gather them on one side first: take 2x from both sides. 3x − 4 = 14.
Undo the subtracting: add 4 to both sides. 3x = 18. Undo the multiplying: divide by 3. x = 6.
Check both sides: 5(6) − 4 = 26, and 2(6) + 14 = 26. Equal, so x = 6 is right.
Solve 2(x − 3) = 10.
There are two ways to do this. Both give the same answer. Divide first: both sides by 2 gives x − 3 = 5, so x = 8. Or multiply out first: 2(x − 3) = 2x − 6, so 2x − 6 = 10, then 2x = 16 and x = 8.
Check: 2(8 − 3) = 2(5) = 10.
Solve x4 + 2 = 5.
Take 2 from both sides: x4 = 3. The x is being divided by 4, and multiplying undoes dividing: multiply both sides by 4. x = 12.
Check: 12 ÷ 4 + 2 = 3 + 2 = 5.
Solve 0.5x + 3 = 7.
Take 3 from both sides: 0.5x = 4. Divide both sides by 0.5: x = 8. (Dividing by one half is the same as doubling, which is a good way to see that 8 is right.)
Check: 0.5(8) + 3 = 4 + 3 = 7.
From words to an equation
A plumber charges $60 to come out, plus $45 an hour. The bill came to $285. How many hours did the job take?
Name the unknown: let h be the number of hours. The bill is the fee plus 45 for each hour: 60 + 45h = 285.
Take 60 from both sides: 45h = 225. Divide by 45: h = 5. The job took 5 hours.
Check with the story: 5 hours at $45 is $225, plus the $60 fee is $285.
Inequalities: the same steps, with one exception
An inequality says one side is bigger than the other, using <, >, ≤, or ≥. Solve it exactly like an equation, with one exception: when you multiply or divide both sides by a negative number, flip the sign of the inequality. (Adding or subtracting anything, or multiplying or dividing by a positive number, leaves the sign alone.)
Solve 4 − 2x > 10.
Take 4 from both sides: −2x > 6. Now divide both sides by −2, and because −2 is negative, flip the sign: x < −3.
Check with a number that should work, like −4: 4 − 2(−4) = 4 + 8 = 12, and 12 > 10. Then a number that should not, like 0: 4 − 0 = 4, and 4 is not greater than 10. The answer x < −3 holds up.
The answer to an inequality is a whole range of numbers, and the test may ask for it on a number line: an open circle at the boundary for < or >, a filled circle for ≤ or ≥, and shading in the direction the sign points.
Two equations at once: a system
A system is two equations with the same two unknowns. Its solution is the one pair (x, y) that makes both equations true. On a graph, each equation is a line, and the solution is the point where the two lines cross. There are two ways to find it without drawing.
Substitution: when one equation already says what y (or x) is, put that expression into the other equation in place of y.
Elimination: when adding the two equations would cancel a variable, add them.
Solve the system: y = 2x − 1 and x + y = 8.
The first equation says what y is, so put 2x − 1 in place of y in the second: x + (2x − 1) = 8. Gather: 3x − 1 = 8. Add 1: 3x = 9. So x = 3.
Now y: y = 2(3) − 1 = 5. The solution is x = 3, y = 5, the point (3, 5) where the two lines cross on the grid.
Check in both: 2(3) − 1 = 5, and 3 + 5 = 8.
Solve the system: x + y = 12 and x − y = 4.
Adding the two equations cancels y, because +y and −y make 0: (x + y) + (x − y) = 12 + 4 gives 2x = 16, so x = 8.
Put 8 into the first equation: 8 + y = 12, so y = 4. Check the second: 8 − 4 = 4. The solution is (8, 4).
One more name for y
The test sometimes writes a line as a function: f(x) = 4x − 3 says the same thing as y = 4x − 3. The letters f(x) are read "f of x," and they stand for the y value that goes with a given x. So f(5) means "put 5 in for x": f(5) = 4(5) − 3 = 17. A question like "for which x is f(x) = 25?" is an equation to solve: 4x − 3 = 25, so 4x = 28 and x = 7.
On test day
The formula sheet is a button on the screen during the whole test. For this material it gives you the slope formula, slope-intercept form, point-slope form, and the Pythagorean theorem, along with the area and volume formulas, the quadratic formula, simple interest, and d = rt. It does not give you standard form Ax + By = C, the midpoint, the distance between two points, or the order of coordinates. You have to remember those four yourself.
The calculator. The test has two parts: a short first part with no calculator, then the rest with the on-screen TI-30XS (you may bring your own TI-30XS to a test center). Slopes are often fractions. The n/d key enters a fraction, and the toggle key switches an answer between a fraction and a decimal, which helps when the answer choices are written one way and your work came out the other. Leave a slope as a fraction unless the choices are decimals.
The question formats. Multiple choice is most of the test. Fill-in-the-blank asks you to type a number: type a fraction with a slash, like 3/4, and a negative with the minus key. Drop-down menus complete a sentence, often the slope or the intercept. Drag-and-drop asks you to build an equation or place labels. Select-an-area asks you to click a point on a grid. Read every question for exactly what it wants: the slope, the y-intercept, the x-intercept, or a point.
Check by putting it back. A point on a line makes the equation true. A solution to an equation makes both sides equal. A solution to a system makes both equations true. Thirty seconds of checking catches most sign mistakes, and sign mistakes are where these questions go wrong: y = 2x − 7 has b = −7; subtracting a negative adds; dividing by a negative flips an inequality.
Time. The whole test is 115 minutes. If a graph question stalls you, flag it, answer the rest, and come back. A question you can check is worth more of your time than a question you are guessing on.
Seventeen questions, four sets
Work each one on paper, type the answer in its box, and check the set. A missed question offers its reasoning with the Why? button, and every set can hand you five more like it, as many times as you want. Your answers stay in this browser until you press Start over.
The grid: points, distance, midpoint
- 1.What are the coordinates of point P?
- 2.In which quadrant is the point (4, −5)?
- 3.How far is it from (−2, −1) to (4, 7)?
- 4.What is the midpoint of the segment from (−6, 2) to (2, 8)?
Slope and intercepts
- 5.What is the slope of the line through (−2, 5) and (4, 1)?
- 6.The line passes through the two marked points. What is its slope?
- 7.Find both intercepts of 2x + 5y = 10.
- 8.What is the slope of the line x = 4?
The three forms
- 9.For the line y = −23x + 4, what are the slope and the y-intercept?
- 10.Write 6x + 3y = 12 in slope-intercept form.
- 11.Write the equation of the line with slope 3 that passes through (2, 1), in slope-intercept form.
- 12.Write the equation of the line through (1, 5) and (3, 1), in slope-intercept form.
Equations and systems
- 13.Solve for x: 5x − 8 = 2x + 13
- 14.Solve for x: 3(x + 4) = 27
- 15.A gym charges $40 to join, plus $25 a month. Someone has paid $215 in all. For how many months?
- 16.Solve the system: y = x + 3 and x + y = 11
- 17.Solve: 5 − 3x ≥ 20
Where to go next
| Set | Questions | Right | Where to go next | Practice |
|---|---|---|---|---|
| 1. The grid: points, distance, midpoint | 1–4 | of 4 | Diagnostic 3, coordinate geometry sectionThe Flyover, family 14 | |
| 2. Slope and intercepts | 5–8 | of 4 | Diagnostic 3, linear equations and slopeThe Flyover, family 13 | |
| 3. The three forms | 9–12 | of 4 | Part 5 above, then Diagnostic 3Use Five more here for practice | |
| 4. Equations and systems | 13–17 | of 5 | The Thing Called x · The Balance · The SharePractice sheet 1 · Diagnostic 3 |
The math pages are at thepeoplesshare.org/ged/math.
The key prints only when it is showing, and it starts on its own page. After you check a set, each question you missed offers its own reasoning with the Why? button.
Answers, with the reasoning
- 1.(−3, 2). Start at the origin. P is 3 squares to the left, so its x is −3, and 2 squares up, so its y is 2. Across first, then up: (−3, 2).
- 2.IV. x is positive (4 to the right) and y is negative (5 down), so the point is in the lower right. Counting counterclockwise from the upper right, that is the fourth quadrant, IV.
- 3.10. Across: 4 − (−2) = 6. Up: 7 − (−1) = 8. Those are the legs of the right triangle, and the distance is the hypotenuse: 6² + 8² = 36 + 64 = 100, and √100 = 10.
- 4.(−2, 5). Average the x values: (−6 + 2) ÷ 2 = −4 ÷ 2 = −2. Average the y values: (2 + 8) ÷ 2 = 10 ÷ 2 = 5. The midpoint is (−2, 5).
- 5.−23. Rise: 1 − 5 = −4. Run: 4 − (−2) = 6. Slope = −46, which reduces to −23. The line falls 2 for every 3 it moves to the right.
- 6.−1. From (0, 5) to (5, 0): the rise is 0 − 5 = −5 and the run is 5 − 0 = 5. Slope = −5 ÷ 5 = −1. The line goes down one square for every square it moves to the right, which is what the picture shows.
- 7.x-intercept 5, the point (5, 0); y-intercept 2, the point (0, 2). For the x-intercept, put 0 in for y: 2x + 5(0) = 10, so 2x = 10 and x = 5. For the y-intercept, put 0 in for x: 2(0) + 5y = 10, so 5y = 10 and y = 2.
- 8.Undefined. Every point on the line x = 4 has x = 4: (4, 0), (4, 1), (4, 2), and so on. It is a vertical line. Between any two of its points the run is 0, and rise over run would mean dividing by 0, which has no answer. A horizontal line, like y = 4, is the one with slope 0.
- 9.Slope −23, y-intercept 4. The equation is in slope-intercept form, y = mx + b. The number multiplying x is the slope, −23; the number added is the y-intercept, 4, the point (0, 4).
- 10.y = −2x + 4. Solve for y. Take 6x from both sides: 3y = −6x + 12. Divide every term by 3: y = −2x + 4. The slope is −2 and the y-intercept is 4. Check with the x-intercept: y = 0 gives 6x = 12, x = 2, and −2(2) + 4 = 0.
- 11.y = 3x − 5. Point-slope form with m = 3 and the point (2, 1): y − 1 = 3(x − 2). Multiply out: y − 1 = 3x − 6. Add 1 to both sides: y = 3x − 5. Check: x = 2 gives 3(2) − 5 = 1.
- 12.y = −2x + 7. Slope first: (1 − 5) over (3 − 1) = −42 = −2. Then point-slope form with (1, 5): y − 5 = −2(x − 1), so y − 5 = −2x + 2, and y = −2x + 7. Check with the other point: x = 3 gives −2(3) + 7 = 1.
- 13.x = 7. Take 2x from both sides: 3x − 8 = 13. Add 8 to both sides: 3x = 21. Divide by 3: x = 7. Check: 5(7) − 8 = 27, and 2(7) + 13 = 27.
- 14.x = 5. Divide both sides by 3 first: x + 4 = 9, so x = 5. Or multiply out first: 3x + 12 = 27, then 3x = 15 and x = 5. Check: 3(5 + 4) = 3(9) = 27.
- 15.7 months. Let m be the months: 40 + 25m = 215. Take 40 from both sides: 25m = 175. Divide by 25: m = 7. Check: 7 months at $25 is $175, plus the $40 to join is $215.
- 16.x = 4, y = 7. The first equation says what y is, so put x + 3 in place of y in the second: x + (x + 3) = 11, so 2x + 3 = 11, 2x = 8, and x = 4. Then y = 4 + 3 = 7. Check the second equation: 4 + 7 = 11.
- 17.x ≤ −5. Take 5 from both sides: −3x ≥ 15. Divide both sides by −3, and because you are dividing by a negative, flip the sign: x ≤ −5. Check with −6, which should work: 5 − 3(−6) = 5 + 18 = 23, and 23 ≥ 20. Then 0, which should not: 5 − 0 = 5, and 5 is not at least 20.