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Learn · Mathematics · Grade 9 · Algebra

First-degree equations and inequalities

Solving ax + b = c step by step, why an inequality flips when you multiply by a negative, and how to draw the solution on a number line.

High school Lesson 3 of 31 Interactive figure Practice

First, the rules

Keep the balance

An equation is a balance: whatever you do to one side you do to the other. An inequality is the same, with one trap.

★ Solving ax + b = c
Examples 1, 2
ax + b = c ⇒ ax = c − b ⇒ x = c − ba
Undo the + b first, then the × a.
Unknowns on both sides
Example 1
Collect the x-terms on one side and the numbers on the other, then divide.
Fractions
Example 2
Multiply every term by the common denominator first — then there are no fractions left.
Special cases
0 · x = 5 no solution · 0 · x = 0 every real number is a solution
★ Inequalities: same moves, one trap
Example 3
Adding or subtracting anything, or multiplying / dividing by a POSITIVE number, keeps the sign. Multiplying or dividing by a NEGATIVE number FLIPS it: −2x < 6 ⇒ x > −3.
Reading the sign
x < 3 open circle at 3, shade left · x ≥ −1 filled circle at −1, shade right
Double inequality
Example 3
−1 < 2x + 3 ≤ 7 ⇒ −4 < 2x ≤ 4 ⇒ −2 < x ≤ 2
Do the same thing to all three parts.

See it

A strip or two rays

The number line below is shared with the absolute-value lesson: |x − a| < c is the strip of points within c of a, |x − a| > c is everything outside. A plain inequality like x > −3 is just one ray — slide a to −3 and look at the right-hand band.

Step by step

Worked examples

1

Unknowns on both sides

collect, then divide

Solve 5x − 7 = 2x + 11.

Ask yourself: Which side will you keep the x on? Move the other x-term there, move the numbers the other way.

Show the solution
  1. Subtract 2x from both sides: 3x − 7 = 11.
  2. Add 7: 3x = 18.
  3. Divide by 3: x = 6. Check: 5 · 6 − 7 = 23 and 2 · 6 + 11 = 23. ✓

Answer x = 6

2

Clearing fractions

common denominator

Solve x3 − x − 14 = 2.

Ask yourself: What number do both denominators divide? Multiply every term by it.

Show the solution
  1. Common denominator 12. Multiply every term by 12: 4x − 3(x − 1) = 24.
  2. Expand the bracket carefully (the minus distributes): 4x − 3x + 3 = 24.
  3. x + 3 = 24 ⇒ x = 21. Check: 21/3 − 20/4 = 7 − 5 = 2. ✓

Answer x = 21

3

An inequality with a negative coefficient

the flip

Solve 3 − 2x ≤ 11 and show the solution on a number line.

Ask yourself: When you divide by −2, what happens to ≤?

Show the solution
  1. Subtract 3: −2x ≤ 8.
  2. Divide by −2 and FLIP the sign: x ≥ −4.
  3. Number line: filled circle at −4 (the equality is included), shade to the right. Test a point: x = 0 gives 3 ≤ 11. ✓

Answer x ≥ −4

Practice: equations and inequalities

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Solve for x — type the number; for inequalities pick the solution set. Watch the sign when the coefficient is negative. Your best score is kept in this browser only.

About this lesson

First-degree (linear) equations and inequalities in one unknown: collecting terms, clearing fractions, the balance rule, equations with no solution or every number as a solution, inequalities and the sign flip when multiplying or dividing by a negative number, double inequalities, and solution sets on a number line. Worked examples and practice with instant checking.

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