Which Inequality Is Shown in the Graph Below? The Three Reads

Here is the graph, exactly as the quiz hands it to you.

Problem. Which inequality is shown in the graph below?

The number line plants a solid dot on 4 and shades right, forever.

The question ‘which inequality is shown in the graph below’ asks you to name the photo in symbols. Read position, dot type, direction — the clues that settle which inequality is shown in the graph below. One question — which inequality is shown in the graph below — three clues.

Answer. Which inequality is shown in the graph below? This one: $x \ge 4$.

The shading runs right, so every working value is greater than 4. The dot is solid, so 4 itself is included — the bar under $\ge$ records that.

Check with test points: on the ray, $5 \ge 4$ is true; off it, $3 \ge 4$ is false. In interval notation the answer reads $[4, \infty)$.

One symbol, two clues — all ‘which inequality is shown in the graph below’ ever asks. Two more graphs below sharpen the same judgment. Read the ray; which inequality is shown in the graph below answers itself.](streamdown:incomplete-link)

Every ‘which inequality is shown in the graph below’ question surrenders to three reads:

1. Read the boundary. The special dot sits on one number; that number enters the inequality.
2. Read the dot. Solid means in ($\le$ or $\ge$); open means out ($<$ or $>$).
3. Read the shading. Right means greater; left means less.

Durell’s 1911 School Algebra settles the symbol: both signs of inequality open toward the greater quantity. Wentworth’s 1898 New School Algebra agrees — the sign points toward the smaller number. Either way, the symbol opens toward the shading.

A quiz asking which inequality is shown in the graph below is really asking which row you are in. That is how to find the inequality represented by the graph. Practice which inequality is shown in the graph below row by row.

Problem. Choose the inequality that represents the following graph — quiz wording for which inequality is shown in the graph below.

Read 1, boundary. The circle sits on 2. Read 2, dot. Open — 2 is excluded, so the symbol is strict. Read 3, shading. Left — less than.

Answer. $x < 2$. Test point 0 on the ray gives $0 < 2$, true; off-ray 5 gives $5 < 2$, false. Another ‘which inequality is shown in the graph below’ item, closed out in three reads — and every graph like this one is read the same way.

Problem. Which inequality is shown in the graph below? The boundary is negative: a solid dot at $-2$, shading right.

Three reads. Boundary $-2$; dot solid, included; shading right, greater.

Answer. $x \ge -2$. Test point 0: $0 \ge -2$, true; point $-5$: $-5 \ge -2$, false.

Where it came from. Solve $-\frac{1}{2}x \le 1$: multiply by $-2$, flip the sign, get $x \ge -2$ — this graph. After any negative divide, recheck direction before answering which inequality is shown in the graph below. Next graph, same question: which inequality is shown in the graph below.

1. Treating an open circle as included. An open circle excludes the boundary. Writing $x \ge 4$ for an open circle at 4 claims 4 works — it does not. Open or closed decides half of which inequality is shown in the graph below.

2. Pointing the sign away from the shading. The sign opens toward the greater quantity, the shaded side. Shading right with $<$ contradicts the graph you just read — and fails any which inequality is shown in the graph below item.

3. Dropping the flip on a negative divide. Example 2 came from $-\frac{1}{2}x \le 1$; clearing the $-\frac{1}{2}$ reverses the symbol. Forget the flip and you shade the wrong half.

4. Misreading the tick marks. Count ticks from a labeled one before naming the boundary. Off-by-one boundaries live on the ticks of every which inequality is shown in the graph below item. Fix these four and which inequality is shown in the graph below stops costing points.