Before you read

Answer first — the lesson below explains it.

Harper needs to earn more than $$17$ from a side job that pays $$2$ per task plus a $$5$ base. The inequality $2 x + 5 > 17$ gives the minimum number of tasks $x$ . What is the solution set?

A
$x > 6$
B
$x > 11$
C
$x < 6$
D
$x \geq 6$

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

Which values of $x$ satisfy $- 3 x \geq 15$ ?

A
$x \geq - 5$
B
$x \leq - 5$
C
$x \geq 5$
D
$x \leq 5$

Check your answer

Answer:

Theory

Inequalities vs. Equations

An inequality is like an equation, but instead of $=$ , it uses:

Symbol	Meaning	Example
$<$	less than	$x < 5$
$>$	greater than	$x > 3$
$\leq$	less than or equal to	$x \leq 7$
$\geq$	greater than or equal to	$x \geq -2$

The solution to an inequality is a range of values, not a single number.

x < 3: open circle, shade left

Solving works exactly like equations, with ONE crucial difference...

Quick check

You just read it. Can you apply it?

If $- 2 x + 7 < 13$ , which of the following must be true?

A
$x < - 3$
B
$x > - 3$
C
$x < 3$
D
$x > 3$

Check your answer

Answer:

Subtract $7$ : $- 2 x < 6$ .
Divide by $- 2$ (flip): $x > - 3$ .

Theory

The Golden Rule: Flip When Multiplying/Dividing by a Negative

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

$-2x > 6 \quad \xrightarrow{\div(-2)} \quad x < -3$

Why? Because multiplying by a negative reverses the order:
- $2 < 5$ is true
- Multiply both by $-1$ : $-2 > -5$ (the sign flips!)

This is the #1 most tested inequality rule on the SAT.

Example 1

Solve $3x + 7 > 22$

Subtract $7$ : $3x > 15$

Divide by $3$ (positive, no flip): $x > 5$

Solution: all values greater than $5$ .

x > 5

Example 2

Solve $-4x + 3 \leq 19$

Subtract $3$ : $-4x \leq 16$

Divide by $-4$ (FLIP the sign): $x \geq -4$

Solution: $x \geq -4$ .

x \geq -4

Example 3

Solve $5 - 2x < 11$

Subtract $5$ : $-2x < 6$

Divide by $-2$ (FLIP): $x > -3$

x > -3

Quick check

You just read it. Can you apply it?

A health guideline recommends that a value satisfies $- 2 x + 8 < 4$ . What values of $x$ meet this requirement?

A
$x = 6$
B
$x = 5$
C
$x = 4$
D
All of the above

Check your answer

Answer:

Solve $- 2 x + 8 < 4$ :
Subtract $8$ : $- 2 x < - 4$ .
Divide by $- 2$ (flip the inequality): $x > 2$ .

Check each option:
A: $x = 6$ : $- 2 (6) + 8 = - 4 < 4$ ✓
B: $x = 5$ : $- 2 (5) + 8 = - 2 < 4$ ✓
C: $x = 4$ : $- 2 (4) + 8 = 0 < 4$ ✓

All three values satisfy $x > 2$ , so the answer is D.

Theory

Graphing on a Number Line

Open circle (○): the endpoint is NOT included ( $<$ or $>$ )
Closed circle (●): the endpoint IS included ( $\leq$ or $\geq$ )
Arrow: shows the direction of all solutions

Examples:
- $x > 3$ : open circle at $3$ , arrow pointing right $\to$
- $x \leq -1$ : closed circle at $-1$ , arrow pointing left $\leftarrow$

Quick check

You just read it. Can you apply it?

A student needs at least 80 points to pass. The score is calculated as $S = 3 x + 20$ , where $x$ is the number of correct answers. What is the minimum number of correct answers needed?

A
$18$
B
$20$
C
$22$
D
$24$

Check your answer

Answer:

$3 x + 20 \geq 80$
$3 x \geq 60$
$x \geq 20$ .
Minimum: $20$ correct answers.

Common Mistakes

Common Mistakes to Avoid

Forgetting to flip the sign when dividing by a negative

$-3x > 12$ $\to$ writing $x > -4$ instead of $x < -4$

ALWAYS flip when you multiply or divide by a negative. No exceptions.

Flipping the sign when subtracting (unnecessary!)

$x - 5 > 3$ $\to$ flipping to $x < 8$

You only flip when multiplying/dividing by a negative. Subtracting does NOT require a flip.

Confusing open and closed circles

Using a closed circle for $x < 3$

<

and

>

use open circles (value not included).

\leq

and

\geq

use closed circles.

Quick check

You just read it. Can you apply it?

If $\frac{x - 3}{2} > 4$ , which of the following must be true?

A
$x > 5$
B
$x > 8$
C
$x > 11$
D
$x \geq 11$

Check your answer

Answer:

Multiply by $2$ : $x - 3 > 8$ .
Add $3$ : $x > 11$ .

Strategy

SAT Strategy Tip

Inequality questions on the SAT often test the sign-flip rule by burying a negative coefficient. Scan for negative coefficients BEFORE solving — if you'll need to divide by a negative at the end, mentally prepare to flip. Also, always check your answer by plugging in a value from your solution set.

After reading

Same question as before. What do you say now?

Harper needs to earn more than $$17$ from a side job that pays $$2$ per task plus a $$5$ base. The inequality $2 x + 5 > 17$ gives the minimum number of tasks $x$ . What is the solution set?

A
$x > 6$
B
$x > 11$
C
$x < 6$
D
$x \geq 6$

Check your answer

Answer:

Subtract $5$ : $2 x > 12$ .
Divide by $2$ : $x > 6$ .

After reading

Same question as before. What do you say now?

Which values of $x$ satisfy $- 3 x \geq 15$ ?

A
$x \geq - 5$
B
$x \leq - 5$
C
$x \geq 5$
D
$x \leq 5$

Check your answer

Answer:

Divide by $- 3$ (flip the sign): $x \leq - 5$ .

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